〜得る・〜得ない
can, is possible; cannot, is impossible
Verb ます-stem + 得る (うる / える) · 得ない (えない)
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得る after a ます-stem means something is possible: 事故は誰にでも起こり得る, "accidents can happen to anyone". It's formal and written, and is about possibility in principle rather than personal ability.
The dictionary form can be read うる or える; the other forms always use え: 得ない (えない), 得ます (えます).
The most common word built on it is あり得ない, "impossible, unthinkable", which is everyday slang for "no way!" among young people: あり得ない!
Other useful combinations: 考え得る, "conceivable"; 起こり得る, "could happen". For a person's ability, use the potential form or できる instead.
Examples
そんなことはあり得ない。
That's impossible.
事故は誰にでも起こり得る。
Accidents can happen to anyone.
考え得る方法はすべて試した。
I've tried every method I can think of.
それは十分あり得る話だ。
That's entirely possible.
彼が犯人だなんて、あり得ないよ。
Him, the culprit? No way.