It first verse of Al-Fatihah is:

بِسْمِ اللهِ الرَّحْمَنِ الرَّحِيمِ

Transliteration as written: Bismi Allahi alrraHmani alrraHiimi

Transliteration as pronounced: Bismillah irroHman irroHiim

Translation: In the name of Allah, the Beneficent, the Merciful.

The first word of this verse is:

بِ

Transliteration: bi

Translation: by, by means of

This is an Arabic word that consists of a single consonant, plus a short vowel. Such single-consonant words are never written on their own, but always joined with the subsequent word. A good number of Arabic prepositions are such single-consonant words. In classical Arabic and the Arabic of the Qur'an, a noun following a preposition should be in the genitive case (ending with –in for indefinite singular nouns, and with –i for definite singular nouns).

The subsequent word is:

اِسْم

Transliteration: ism

Translation: name

When the word "by" is placed before a noun, this word takes the genitive case, which in classical Arabic is marked by a suffixed short "i". While in modern colloquial Arabic, this "i" may be neither written nor pronounced, it is used historically in the Qur'an.

Thus:

بِسْمِ

Transliteration: bismi

Translation: In the name

In Arabic, genitive constructions are a very common feature.  Basically, two nouns are placed next to each other.  In our case, the two nouns that stand next to each other are "name" and "Allah". In a genitive construction, the first noun can be any grammatical case, but it never takes an article. The second noun is in the genitive, which in classical Arabic has a suffixed short "i", which, however, in modern colloquial Arabic may be neither written nor spoken.

Thus, we can form:

اِسْم اللهِ

Transliteration: ism Allahi

Translation: The name of Allah

Remember: the definite article "al" is never written for the first word of a genitive construction.

When we add the Arabic word for "by" (bi) in front of "name", the word "name" itself becomes a genitive. As a single-consonant word, the Arabic word "bi" is always joined with the next word, in this case "ism".

Thus:

بِسْمِ اللهِ

Transliteration as written: bismi Allahi

Translation: In the name of Allah

However, Arabic isn't always spoken as it is written. Sound assimilation plays a major role. Already, the joining of "bi" and "ism" doesn't render "biismi" but "bismi".

The word 'Allah' starts with the same two letters as the Arabic definite article:

ال

Transliteration: al

Translation: the

Actually, the definite article is an integral part both of the word and the meaning of "Allah" = the God, meaning: the only God. 

Thus, the same phonetic rules that apply to the definite Arabic article, al, also apply to the word Allah.

If the word preceding the definite article al or the word Allah ends with a vowel, then the "a" of al, or the initial "a" of Allah is omitted in the spoken language, even though the alif is still written.

But not only does the final vowel of the preceding word cause the "a" of al to be lost. The final vowel of the preceding word also phonetically connects the preceding word with the word beginning with al (or the word Allah) and phonetically even moves from the end of the preceding word to become the first sound of the word beginning with the definite article al (or the first sound of the word Allah).

Thus:

بِسْمِ  اللهِِ

Transliteration as spoken: bismillahi

Translation: In the name of Allah

Some teachers may insert a hyphen between the two words: bism-illahi or bismi-llahi, to indicate that the sound is formed of two words; bism-illahi is closer to how the combination is spoken. On the other hand, bismillahi is actually not just two words, but three: bi-smi-llahi = by name Allah.

بِسْمِ اللهِ الرَّحْمَنِ الرَّحِيمِ

In the first verse of Al-Fatihah, two adjectives follow the noun Allah, the adjectives for "beneficent" and "merciful". In Arabic, adjectives are much closer to nouns than in English, Furthermore, adjectives always follow nouns. To understand the noun-like nature of Arabic adjectives, it is best to translate according to the following pattern:

English: a beautiful girl

Arabic-like structure: a girl a beautiful(-one)

English: the young man

Arabic-like structure: the man, the young(-one)

English-like structure: In the name of the beneficent and merciful Allah

Arabic structure: In the name of Allah, the beneficent(-one), the merciful(-one)

The post-positioned Arabic adjectives always have the same grammatical form as the noun they describe. Therefore, as the word Allahi is definite singular genitive, both adjectives have to be the same.

The first adjective, as it is included in dictionaries:

الرَّحْمَن

Transliteration as written: alrraHman

Translation: the Merciful

However, again because of sound assimilation, the word is spoken slightly differently. The l of the definite article al is omitted, and the first consonant of the actual adjective is duplicated, indicated in writing by the use of the shadda:

ّ

This rule applies to about half of all Arabic consonants. The consonants to which this rule applies are commonly names "Sun" letters. The Sun letters are:

ت ث د ذ ر ز س ش ص ض ط ظ ل ن

For all the above, if the definite article al is placed before a noun or adjective beginning with that letter, the l of the article al is omitted, and the initial consonant is duplicated.

The second adjective, as it is included in dictionaries;

رَحِمَ

Transliteration as written: raHima

Translation: to have merci

In Arabic, any adjective always follows the grammatical case of the noun it describes. Thus, because Allahi is genitive and has the genitive i, the subsequent adjective(s) must also be in the genitive and bear the i.

Then, we have the rules of sound changes (not written) and sound assimilation.

First, a sound change. Arabic, of course, does not write a letter o, However, when a word contains an emphatic consonant. Such as the emphatic H (middle of the tongue depressed to the base of the oral cavity), the sound of the a next to the emphatic letter will sound like an open o.

And then, as explained already further above, there are the rules of sound assimilation for the article al. The last vowel of a word preceding a definite article tends to replace the a of al in the subsequent word.

So, the first verse is grammatically explained:

بِسْمِ اللهِ الرَّحْمَنِ الرَّحِيمِ

Transliteration as written: Bismi Allahi alrraHmani alrraHiimi

Transliteration as pronounced: Bismillah irroHman irroHiim

Translation: In the name of Allah, the Beneficent, the Merciful.

It second verse of Al-Fatihah is:

الْحَمْدُ لِلَّهِ رَبِّ الْعَالَمِينَ

Transliteration as written:  Alhamdu lillahi rabbi alAAalameena

Translation: Praise be to Allah, Lord of the Worlds

If we consult a modern Arabic dictionary, we will find the following entry:

حَمْد

Transliteration: hamd

Translation: praise (indefinite noun)

To turn this into a definite noun, we ad the article "al":

الْحَمْد

Transliteration: al-hamd

Translation: the praise (singular)

For modern colloquial Arabic, this would be all that is needed. However, classical Arabic, and the Arabic of the Qur'an, uses grammatical case endings.

Indefinite singular nominative (English sample sentence: Praise is welcome.): hamdun

Indefinite singular accusative (English sample sentence: The teacher gave praise.): hamdan

Indefinite singular genitive (English sample sentence: The sound of praise was loud.): hamdin

Definite singular nominative (English sample sentence: The praise is welcome.): al-hamdu

Definite singular accusative (English sample sentence: The teacher gave the praise.): al-hamda

Definite singular genitive (English sample sentence: The sound of the praise was loud.): al-hamdi

So, what we have in the second verse of Al-Fatihah is "praise" in the singular nominative:

الْحَمْدُ

Transliteration: al-hamdu

Translation: the praise

The next word is actually again two words:

لِلَّهِ

Transliteration: lillahi

Translation: for Allah

It is a contraction of the preposition li = for and Allah.

لِ + الله = لِلَّهِ|

It third verse of Al-Fatihah is:

الرَّحْمَنِ الرَّحِيمِ

Transliteration as written: Alrrahmani alrraheemi

Translation: The Beneficent, the Merciful.

It fourth verse of Al-Fatihah is:

مَالِكِ يَوْمِ الدِّينِ

Transliteration: Maliki yawmi alddeeni

Translation: Master of the Day of Judgment.

It fifth verse of Al-Fatihah is:

إِيَّاكَ نَعْبُدُ وَإِيَّاكَ نَسْتَعِينُ

Transliteration: Iyyaka naAAbudu waiyyaka nastaAAeenu

Translation: Thee do we serve and Thee do we beseech for help.

It sixth verse of Al-Fatihah is:

اهْدِنَا الصِّرَاطَ المُسْتَقِيمَ

Transliteration as written: Ihdina alssirata almustaqeema

Translation: Keep us on the right path.

The second word in this verse is included in dictionaries as:

صِرَاط

Transliteration as written: SiraaT

Transliteration as spoken: SirooT (the emphatic T at the end causes the preceding aa to sound like an open oo)

Translation: path

Because the word occurs with the definite article al, and because the first letter of the noun is a sun letter, the sound is assimilated, and because the word occurs in the accusative, an a is suffixed:

الصِّرَاطَ

Transliteration as written: alSSiraaTa

Transliteration as spoken: aSSirooTo

The third word of this verse, as included in dictionaries, is:

مُسْتَقِيم

Transliteration: mustaqiim

Translation: straight

Because this adjective describes a preceding noun in the accusative with a definite article, the adjective, too, must be in the accusative and take the definite article. Together, the two are:

الصِّرَاطَ المُسْتَقِيمَ

Transliteration as written: alSSiraaTa almustaqiima

Translation: the straight path

Apart from the sound assimilation that affects alSSiraaTa, there is a further sound assimilation that occurs between the two words. Because the previous word ends with a vowel, the a of the al in almustaqiima is dropped and replaced by the ending vowel of the previous word. This vowel may be written as an a, but because it occurs next to an emphatic T, the sound changes to that of an open o. And because of ease of pronunciation, both the o and subsequent l appear to stick closer to the previous noun than the mustaqiima.

Therefore, we have the following transliteration as spoken: Ihdina SSirooTol mustaqiim

It seventh verse of Al-Fatihah is:

صِرَاطَ الَّذِينَ أَنْعَمْتَ عَلَيْهِمْ غَيْرِ المَغْضُوبِ عَلَيْهِمْ وَلاَ الضَّالِّينَ

Transliteration: Sirata allatheena anAAamta AAalayhim ghayri almaghdoobi AAalayhim wala alddalleena

Translation: The path of those upon whom Thou hast bestowed favors. Not (the path) of those upon whom Thy wrath is brought down, nor of those who go astray.

This verse includes several linguistic options of Arabic negations.

غَيْر

Transliteration: ghair

Translation: not, other than, not the same

A common derived word is:

غَيْرَة

Transliteration: ghaira

Translation: enthusiasm

The Arabic word has also made it into the Malay / Indonesian vocabulary, spelled gairah, with the same meaning, Unfortunately, there is no logical way to know the  meaning of derived Arabic words. They have to be learned just as independent words,