This app was mentioned in 4 comments, with an average of 1.75 upvotes
I used this app for awhile but then started using ScoreGeek which I prefer. It has advanced API options to fully create your own game scoring guide. It also has BBG implementations and I find it an all around better score app.
Although BoardGameGeek has an app, it's not that great and there are a couple of better ones:
ScoreGeek is what I currently use. Aside from logging plays, it has score calculators for tons of games that otherwise have tedious end-of-game scorekeeping math. It also tracks stats for each game and each player, and has achievements for things like highest score recorded for a particular game, player's personal bet scores, etc.
ScorePal is the other alternative, and although I started using ScoreGeek instead because it had more features, Scorepal is being updated quite often and has added most of the features I was looking for. The interface is also better, so I may end up switching over to it soon.
I use scoregeek. It has a database of most games, but you can create and share custom scoring cards for any that don't exist. It will also automatically post plays to you and your group members' BGG accounts.
You can be as meticulous or lazy with it as you want, I typically just do a win/lose rather than record all of the details (faction used, points for science, points for pudding/sashimi/each type of sushi etc. etc.) worth the $4 for me.
I use ScoreGeek and I find it does everything I want. It does BGG integration too.