First things first:
Aaron's webpage has a set of very nice cheat sheets that go over the process in detail.
Summoning is relatively straightforward. First, choose your spirit type and force. What types of spirits you can summon is based on your tradition. The GM rolls dice equal to the force of the spirit vs. your conjuring pool, consisting of your Conjuring + Magic. For each net success you have, you have one service from the spirit. Total up the spirit's successes (not net successes, total successes) and multiply by two. That's the Drain Value you're going to resist, using your Willpower + drain attribute (as determined by your Tradition).
That's pretty much the basics of it. There are other steps, bonuses that may or may not apply, and the like; but that's the overview you were asking about.
Edit: What each spirit can do depends on the spirit type and force. For every 3 full points of Force, you can select one optional power from the spirit's list. You pick that when you summon the spirit. So, a Force 3 spirit has one optional power, a force 6 has two, and so on and so forth.
Any spirit can use its powers in combat if ordered to, it's just that some have better attack powers than others. Elemental Attack tends to be better than straight unarmed combat, for example. They all have Immunity to Normal Weapons, though, just for materializing; that gives them hardened armor equal to double Force. So, a Force 6 spirit is pretty much impervious to any non-heavy weapon. There are ways of bypassing this, but you won't need to worry about them for a while.