What service can be used to save and replay/analyze a chess match? I've been starting to write down my chess matches in a scorebook, but would like to find a site where I can transcribe the match and then replay move by move by game to help me analyze what I did well and where I made mistakes.

I remember accidentally finding this kind of a SAAS a while ago, but I never bookmarked it, and don't even know what to search for anymore.

Can anyone give me some pointers what to search for, or recommend a site that can provide this kind of service?
For an online tool, lichess is good. You can enter your game and have it analysed by a computer engine. 

The analysed game will have annotations in plain language ("mistake/blunder/inaccuracy") and of course contain the computer evaluation score after each move. 

Once you have a game analysed on lichess, they have another good feature called: "Learn from your mistakes". Here it will ask you at every mistake/blunder to find a better move.

Another feature which might help you is called "chess insights", basically statistics of your games (e.g. which opening did you score best, etc).

You can do much the same (and more) offline using a chess database software. Scid is a popular free one, chessbase a popular non-free one. However as a beginner you might find the lichess features sufficient and more easy to use. In any case you can import/export between lichess and chess databases.

Particularly for a beginner it is difficult to properly analyze a game. Computers will tell you about tactical mistakes, but finding problems in strategy/positional play using computers will be just about impossible on your own. Therefore I would suggest to find a stronger player look over your games and give you tips. Ideally you'd join a chess club, but if that's not an option you could also post your games in online forums. 

Another way to improve is to read or watch commented games, particularly those in the openings you play.