I - CONSTRUCTION
1 - Godan verbs.
The potential form of the Godan verbs is constructed by changing the /u/ sound to /e/ and adding る :
読む → 読める ; 帰る → 帰れる ; 買う → 買える
It's actually a contraction of 読まれる, 帰られる and 買われる which are now only used as the passive form (I think).
2 - Ichidan verbs.
The potential form of the Ichidan verbs is constructed like the passive form by adding られる to the stem (verb at it's simple form minus る) :
食べる → 食べられる ; 忘れる → 忘れられる
The contracted forms 食べれる and 忘れれる, though widely used, is not considered gramaticaly correct.
3 - Irregular verbs.
Watchout for ある which have a different kind of potential form.
来る → 来られる ; する → 出来る{できる}
II - Meaning
1 - Potential.
忘れられない人。
Someone unforgettable.
世界大戦が起これる。
A world war can happen.
2 - Capacity
犬は泳げる。
Dogs know how to swim.
ついに私は読める!
I'm finally able to read.
III - Expressions with similar meaning.
1 - ことができる.
Verb (simple form) + ことができる has the same meaning as the potential form (potential + capacity).
飛ぶことができると思います。
I believe I can fly.
2 - 得る{える} and 得る{うる}.
Verb stem + える (literary) or うる (more literary) has the meaning of potential but not capacity. It's also the only potential form of ある.
そんな馬鹿な事はありえないよ。
Something that stupid can't exist!
ガラスは圧力で割れ得る。
Glass can break under pressure.
IV - Regarding the particles and your examples.
The basic sentence is (neutral and polite) :
私は本を読む。 私は本を読みます。
I read a book.
The passive form isn't 本が読みます* but :
私に(は)本が読まれる。 私に(は)本が読まれます。
The book is read by me. (unnatural)
The gramatical subject of the potential verb in japanese isn't the one being able to do but the thing that can be done :
私(に)は本が読める。私(に)は本が読めます。
I can read a book. (Litteraly) The book can be read by me.
Due to the influence of the occidental languages, and mainly english, you can also find が (insisting on the object "the book") replaced by を (insisting on the action "to read").
The difference between が and を with the potential form of a verb.
、in Japanese. It has a space built-in.