I really wish I could give credit to the author of the original post, but the user has deactivated their account...
Please note that because imperfect formatting really annoys me, the following text is no longer a perfect copy of the original. The meaning remains unchanged, but I'm making some edits to the structure and style, in the hope of making it more readable. (I'm sorry, it's going to look awful before it starts looking better.) After some hesitation, this will include a contraction that I have observed but was not mentioned by the original author, and some re/written explanations of patterns demonstrated in the original text.
Elle
Elle usually contracts to e, but becomes l' in front of est.
Elles also contracts to e, but becomes e'z' in front of a vowel.
Elle m'énerve = E m'énerve.
Elles sont belles = E sont belles.
Elle est belle, elle = L'est belle, elle.
Elles ecoutent la radio = E'z'ecountent la radio.
Il
Il becomes y when followed by a consonant and l' when followed by a vowel.
Ils becomes y when followed by a consonant but y z' or z' when followed by a vowel.
Il va venir = Y va v'nir.
Il est bizarre, lui = L'est bizarre, lui.
Ils vont au magasin = Y vont au magasin.
Ils ont de la chance = Y z'ont d'la chance/ Z'ont d'la chance.
Il (Impersonal Pronoun)
Il can be dropped before conjugated forms of falloir (eg (Il) faut, (Il) faudrait, (Il) fallait):
Il faut que je parte = faut que je parte/faut qu'je parte/faut que j'parte.
Il can also be dropped from the expression il y a (and its variants):
Il y a = Y a OR Ya.
Il y a du monde ici = Y a du monde ici.
Il y aurait = Y aurait.
Il y avait = Y avait.
Il y a eu = y a eu.
-re
The ending -re may also get dropped sometimes for easy flow. This also applies to the "re" verbs.
Le pauvre chat = Le pauv'chat.
Un, deux, trois, quatre, cinq = Un, deux, trois, quat', cinq.
Notre maison = Not'maison.
Votre mere = Vot'mere.
Il va mettre son cahier sur la table = Il va mett'son cachier sur la table.
N.B. The re sound can only be dropped when it ends a word, not when it's in the middle:
Apprendre, comprendre etc must retain the middle re sound.
Miscellaneous (ie I haven't really reorganised this yet!)
Celui becomes c'ui (pronounced like "oui"):
Tu connais celui-la? = Tu connais c'ui-la?
C'est un/une becomes c't'un/une:
C'est un bon ami = C't'un bon ami.
C'est une bonne amie = C't'une bonne amie.
Parce que becomes pasque:
Elle sourit parce qu'elle est contente = Elle sourit pasqu'elle est contente.
Petit/petite becomes P'tit/p'tite:
Il est petit = Il est p'tit.
Elle est petite= Elle est p'tite.
Peut-etre becomes p't'et:
Peut-etre qu'il est malade = P't'et'qu'il est malade.
Quelque becomes queque:
Quelque part = Queque part.
Tu veux quelque chose à manger ? = Tu veux queque chose à manger ?
Qui becomes qu' when followed by a vowel. Don't do this when qui begins a sentence.
Aujourd'hui, il a rencontré un homme qui a cent ans = Aujourd'hui, il a rencontré un homme qu'a cent ans!
S'il te plait becomes s'te plait:
Passe-moi le beurre, s'il te plait = Passe-moi l'beurre, s'te plait.
Tu es becomes t'es:
Tu es nulle ! = T'es nulle ! (Very uplifting, I know; sorry, it was the only example I could recall off the top of my head.)
/ B1 : 
/ A2 : 