The most common current usage of this word appears to be political: you endorse this or that candidate, or – perhaps moreso – some other person or organization does. It’s like they’re giving the candidate a jersey with their team name on the back. (If you or I endorse a candidate, of course, it’s not really a team jersey so much as just a pat on the back.) It probably also comes with something like a blank cheque, at least as far as action is concerned, and perhaps as far as money is concerned, too – or at the very least funds to use ad libitum.
We speak of endorsing ideas and similar things, too, of course; the usage transfers easily. But there is another use that’s not just common but standard: what you do to a cheque to make it negotiable. If the cheque is made out to you, it’s only good for you; once you endorse it – sign it on the back – it can be negotiated by your bank, or, for that matter, redeemed by someone else.
That’s actually the sense from which the political sense comes. The source of this word is Latin indorsare, from in “upon” and dorsum “back”, meaning “write on the back [of whatever]”. The French makes it a bit more transparent: endosser, with dos – “back” – right there in the middle.
We don’t use cheques and similar personal means of transferring funds as much as we used to, so let me tell you why endorsing a cheque is like endorsing a candidate: it at least used to be the case that if you endorsed a promissory note, cheque, banknote, or whatever, you were taking on responsibility for its being paid. The buck stopped at you, so to speak. (This is not the origin of that phrase, by the way; passing the buck comes from passing the deal around in a poker game – the dealer was originally indicated by possession of a buckhorn knife, it seems.) So by endorsing someone or something you are giving your word that he, she, or it is good.
Endorse is a warm kind of word, especially at the start: it has that nice nasal, a good echo of endorphin, and the air kiss of the /or/. After that it cools a bit with the air of the /s/, but it also gets an echo of horse. (End horse? Perhaps, but you hope it won’t be a horse’s end.) And if your candidate is the horse you’re backing, well, it’s at least a pat on the back, and it may well be your name on the back, too – of the jersey, or of a cheque.