This is the way it works for many companies:
In theory every shareholder gets a vote. One vote per share. Now here's the trick. In order to vote you have to come to a shareholder meeting, which is often deliberately scheduled on a Wednesday somewhere that's expensive to get to or stay at. This is done deliberately to make it a pain for those people who only own a few shares to vote for things.
Now what shareholders can do is let someone vote for them. That person is their "proxy" or representative. Almost always these proxies don't just represent one person, but rather an entire faction. So, you have each of the factions competing to get shareholders allow them to act on their behalf. Now it sounds like a representative democracy, but what really happens is the big players game the system. That's why shareholder meetings are made as annoying to attend as possible, and there's nothing like a mail in ballot.
Theory is shareholders (vote on) -> board members (select)-> company CEO/President
Reality is shareholders (give power to)-> proxies (vote on) -> board members (select)-> company CEO/President
Knowing this a proxy fight is merely trying to sway shareholders and proxies to your way of thinking.