To Die Out
To Come To An End
intransitive verb, ichidan verb
Because the hiragana ends with an う sound, you know this word is a verb. The kanji itself means die out so the verb vocab version is to die out or to come to an end.
絶える is mostly used for when something that was ongoing comes to an end, like breath stopping, laughter fading away, a supply being cut off, or foot traffic dying down. You know this because when something dies out, nothing is left but thin air (える).
The reading can come from 絶つ, so as long as you know that word you know this one as well.
電気や水が絶えることなく供給されているのはとてもありがたいことです。
We are very grateful for the uninterrupted supply of electricity and water.
悲しいけど、コウイチの命はここで絶えてしまったんだ。
It's sad, but Koichi's life has ended here.
トーフグメンバーが笑い茸と呼ばれるキノコを食べてからというもの、トーフグのオフィスでは、笑いが絶えることはない。
Since Tofugu members ate a mushroom called "panaeolus papilionaceus," there has been no end to the laughter in the Tofugu office.