Score Voting is simply any voting system where votes are cast as a "score" in some numerical range. This range can be as simple as the discrete range of 0 or 1 (which is Approval Voting), all the way to a full continuous scale such as the rational numbers. The scale can allow negative scores.
In score voting you simply add up the scores and the candidate with the highest sum is the winner.
The only downside of Score Voting that needs to be addressed is it allows "score extremism", where people give every candidate they even slightly like the maximum score and all others the minimum score, even if that isn't an honest scoring. However score voting systems always satisfy the monotonicity criterion, since it's never rational for a voter to score a candidate they like less above one they like more.
Weighted Score Voting is where voting weights are placed on the election as a whole, and then all the scores are multiplied by that weight. This has the effect of gauging how much concern a single voter has for the results of that election specifically.
You can read more about the theoretical aspects here.