The long answer tells you where the parties do show up in the process. It is a lot like physics: You can spend your life perfectly happily at the Newtonian level and never notice a discrepancy.

(By the way we don't use the term "Federal" for elections, we say "national".)

1) To determine who will be on the ballot for president for each political party, we have "primary" elections. These are handled separately by each state in the union. The exact manner of choosing is decided by each state, so there is some variation.

I think California is somewhat typical, so I will tell you how we do it here: all voters are registered by party. I am a registered Democrat, for instance. This means I will choose between the Democratic presidential hopefuls. In '04 I had to choose between Kerry, Clark, Dean, Edwards and the other ones I'm forgetting. Many states have laws that permit write-in candidates (in case you feel like wasting your vote).

Note that these primaries are not all held on the same day, but instead are spread out over a months-long "primary season".

2) At this point you can quit hurting your head if you just assume that somehow, proportional to how much support the candidates get in the primaries, a candidate for president is chosen for each party (skip down to #3). You want to hurt your head a bit, read on. The parties hold conventions where delegates from each state attend in numbers proportional to the Democrats voting in the primaries in each state. They poll these delegates to determine who has the most support and wins the nomination. In this process too, the states determine the rules governing how each state's delegates must vote. Assume for the sake of simplicity that they vote proportionally to the number of votes that the candidates received in their state's primaries.

3) Somewhere along the line (and with much hoopla), a vice-presidential candidate is announced. The vice-presidential candidate is flat-out chosen rather than voted on. At this point, you have a "ticket" (prez nominee plus vp nominee).

4) Then the real campaign season starts. Usually there are debates between the tickets. Usually there is some controversy over excluding minor parties (those other than Republican and Democrat) nominees from the debates.

5) In November you have the actual national election. You are able to choose between the parties' tickets. In '04 we had to choose between Bush/Cheney, Kerry/Edwards, Ralph Nadar/??, err the Libertarians (whoever they fielded), etc... I think, but can't remember for sure if we have the option to write-in a choice if you want to waste your vote. Anyone recall if there's the possibility of write-in at the national level?

6) Now at the Newtonian level you can assume that whichever ticket gets the most votes wins, and you would be right 99% of the time. If you want to hurt your head, then we have to talk about the electoral college.

Each state, proportionally to their populations gets a certain number of electoral college votes. Depending on which ticket wins in a given state, all of that state's electoral college votes goes to that particular ticket. The ticket with the most electoral college votes wins. And yes it is possible to win the presidency (most electoral college votes) but lose the "popular vote" (simple total of all votes cast in the country). It is one of those spots where Einsteinian physics starts to glare.