Q:

Database design schema for a multi event competition, step 2 will be tracking the same data over multiple years

I am new to databases and I can't wrap my head around the schema I need. 
I have a table for competitors, and a table for events. Each competitor will compete in each of the 8 events. I believe I need a bridge table, and I have looked at the student/class model. I believe my problem lies in the fact that I am trying to place each of the 10 scores inside the event table, but I don't know how to make the association between them.
Here is where I am at with my model:
Click here for schema

A:

From what you describe, the Score1 to Score10 are pieces of information that relate to a specific competitor competing in an event, so those would rightfully belong in the bridge table.
You could also choose to have each score as a separate row in that bridge table, or to keep them all on a single row. You might choose to do the latter if you know that there will always be ten scores each time a competitor competes in an event.
So, your bridge table might look like this:
CREATE TABLE EventCompetitor (
    EventID int NOT NULL,
    CompetitorID int NOT NULL,
    Round int NOT NULL,
    Score int NULL,
    CONSTRAINT fEventID_EventCompetitor FOREIGN KEY (EventID) REFERENCES Event(EventID),
    CONSTRAINT fCompetitorID_EventCompetitor FOREIGN KEY (CompetitorID) REFERENCES Competitor(CompetitorID),
    CONSTRAINT pEventID_CompetitorID_Round_EventCompetitor PRIMARY KEY CLUSTERED (EventID, CompetitorID, Round));

So this way there would be ten rows to represent the ten scores that a competitor gets in the event and to find out the final score you just add all the scores:
SELECT EventID, CompetitorID, TotalScore = SUM(Score)
  FROM EventCompetitor;

The other option would be for the bridge table to look like this:
CREATE TABLE EventCompetitor (
    EventID int NOT NULL,
    CompetitorID int NOT NULL,
    Score01 int NULL,
    Score02 int NULL,
    Score03 int NULL,
    ...
    CONSTRAINT pEventID_CompetitorID_EventCompetitor PRIMARY KEY CLUSTERED (EventID, CompetitorID));

And your query to get the total score would be like this:
SELECT EventID, CompetitorID,
       TotalScore = ISNULL(Score01, 0) + ISNULL(Score02, 0) + ISNULL(Score03, 0) + ...
  FROM EventCompetitor;

