In Japanese, the ability to do a certain action is expressed by conjugating the verb rather than adding a word such as the words "can" or "able to" in the case of English. All verbs conjugated into the potential form become a ru-verb.
Once again, the conjugation rules can be split into three major groups: ru-verbs, u-verbs, and exception verbs. However, the potential form of the verb 「する」 (meaning "to do") is a special exception because it becomes a completely different verb: 「できる」 (出来る)
※Remember that all potential verbs become ru-verbs.
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It is also possible to just add 「れる」 instead of the full 「られる」 for ru-verbs. For example, 「食べる」 becomes 「食べれる」 instead of 「食べられる」. I suggest learning the official 「られる」 conjugation first because laziness can be a hard habit to break and the shorter version, though common, is considered to be slang.
The potential form indicates that something is possible but no actual action is actually taken. While the potential form remains a verb, since it is describing the state of something you must not use the direct object 「を」 as you have done with regular verbs. For example the following sentences are incorrect.
Here are the corrected versions:
There are two verbs 「見える」 and 「聞こえる」 that mean that something is visible and audible, respectively. When you want to say that you can see or hear something, you'll want to use these verbs. If however, you wanted to say that you were given the opportunity to see or hear something, you would use the regular potential form. However, in this case, it is more common to use the type of expression as seen in example 3.
You can see that example 3 uses the generic noun for an event to say literally, "The event of seeing movie was able to be done." which essentially means the same thing as 「見られる」. You can also just use generic noun substitution to substitute for 「こと」.
Here's some more examples using 「聞く」, can you tell the difference? Notice that 「聞こえる」 always means "audible" and never "able to ask".
You can say that something has a possibility of existing by combining 「ある」 and the verb 「得る」 to produce 「あり得る」. This essentially means 「あることができる」 except that nobody actually says that, they just use 「あり得る」. This verb is very curious in that it can be read as either 「ありうる」 or 「ありえる」, however; all the other conjugations such as 「ありえない」、「ありえた」、and 「ありえなかった」 only have one possible reading using 「え」.