Potential Form

Can do verb
Ichidan Verbs: Verb Stem + られる
Godan Verbs: Verb Stem + える
Irregular: する→できる ・くる→来られる

The potential form of a verb simply means “is able”, “is possible”, or “can”
The potential form of a verb is a state. As such they use が almost all of the time, especially in shorter sentences.

Examples

• 日本語の本が読めます – I can read Japanese books
• 日本語の本が読めました – I was able to read a Japanese book. (I could at that time, might still be able)
• 日本語が話せます – I can speak Japanese.
• 山に登れた – I was able to climb the mountain (to the top).

を

を can sometimes be used in place of が; shifting focus to the action (turns it back into a verb); However Some people say this is technically incorrect. を tends to be more common with “can’t” or “could” type statements. More space between the object and the verb seems to increase the acceptability of を.
You may not use を in front of できる. 

• 弁当をどこで買えますか？ – Where can I buy a lunch?
• いつでもぬいぐるみを作れます – I can make a stuffed animal anytime. 
• 仕事をいつでもやめられます – I can quit my job at anytime.
• いつでも仕事がやめられます – The job can (be) quit at anytime. 
• 人を助けるのをやめられない – I can’t stop helping people. 
• 私は砂糖菓子を食べるのをやめられない。- I can’t stop eating sweets. (Pulled from weblio)
• 彼はまだ食事を食べられない。- He can’t eat food yet.  (Pulled from weblio English)

Dropping ら

Ichidan verbs can drop ら from られる to shorten the sound. This only applies to the potential form られる. This is quite common. 

• 食べられる→食べれる
• 温められる→温めれる

ことができる

Doing X is possible
Dictionary form verb + ことができる

Another way to express the potentiality of a verb is with ことができる。
ことができる is the bare minimum “physically capable” meaning of can. It can also be used as a more formal way of communicating potential. 
More simply ことができる = Possible, Potential form ＝ Can. 

• 手伝うことができる – I could help you.
• 手伝うことができない – I couldn’t help you (I have no hands)
• 手伝える – I can help you. 
• 手伝えない – I can’t help you. (I’m busy)

Other examples:

• 美味しい料理を食べることができました – We were able to eat good food (probably a thank you).
• 君のおかげで楽しめた – We were able to enjoy it thanks to you. （We enjoyed it because of you)
• 君のおかげで楽しむことができた – It was possible to enjoy it because you were there.

Reference

Potential or Passive?

Ichidan verbs look the same in potential form and passive form; So we must rely on particles and context to tell them apart. Unfortunately, there’s only two rules I can think of that are dead giveaways:

Passive:

• Might have an enabler/partner marked with に. 

Potential:

• Can be abbreviated from られる to れる

There’s a lengthy discussion about this here. 

In cases where it is ambiguous, ことができる can be used to help distinguish potential from passive by the speaker/writer.