[出]{で}[会]{あ}う is a verb that means to come across or in some contexts to meet (but not in the sense of "have a meeting with"). Thus it's basic meaning is to chance upon something.
[出]{で}[会]{あ}い is two things -- and it's not a coincidence.
[出]{で}[会]{あ}い is what is often-called the "masu-stem" in English or 連用形 in Japanese. This is a productive form which for 五段 / type i verbs like those that end in う changes to い. (For 一段 / type II verbs, the final sounds disappears -- 食べる -> 食べ)
Thus, one use is when you join 出会う to the helping verb ます to make it more polite. = 出会い + ます = 出会います
The other thing that happens is that for many verbs the masu-stem / 連用形 has become a noun as well with a meaning like an -ing noun in English.
So 出会い can also mean "meeting" (under the meaning to chance upon something or encounter it).
Similar production can be seen elsewhere:
遊ぶ -> 遊び
話す -> 話し
飲む -> 飲み
食べる -> 食べ
切る -> 切り
My sense is that this not universally productive (meaning I don't think every single verbs' 連用形 will be a known and usuable noun).