I learned this kanji in Tokushima at a jazz bar. It was a bar in one of those buildings with several different bars packed onto each floor. A man got on the elevator just before I did and held the door for me. Turns out he was going to the same bar I was. There was hardly anyone in the bar – only a few others who turned out to be the band that was playing that night – so we talked for a while, and he told me how surprised he was that I ended up going into the same bar he did. It also happened that he owned a bar himself, so he invited me to stop by and booze a bit the next night. He said that the random chance that we went into the same bar was 何かの縁, which I like to translate as “some kind of connection/fate.” The usage is with 〜ある, so 何かの縁がある.
It has a host of other meanings – relation, connection, ties, fate, destiny, marriage, conjugal relations, chance. A very multifaceted kanji. It has the thread thingy on the left and reminds me of 緑（みどり） and 豚（ぶた）.
One of the simplest usages is 縁を結ぶ (えんをむすぶ) – to form a connection with.
The 新明解 (しんめいかい), a famous Japanese dictionary, lists the fate definition as the first one and also notes that it comes from Buddhist philosophy. It gives two examples:
前世の縁（ぜんせのえん）- a connection with/to a past life
妙な縁で彼に会う（みょうなえんでかれにあう）- (I/he/she/they/we) meet him by a strange turn of fate