If I had to pick the most famous philosopher from the Middle Ages it'd probably Thomas Aquinas, but to be honest I like Anselm of Canterbury the best. Generally medieval philosophers go hand-in-hand with theology because their topics are usually what is the nature of the Trinity or something along those lines. To do this they rely heavily on the Bible, which I dislike because using revelation in philosophy doesn't seem like the best method since your point is "because God says so." Aquinas doesn't have a single argument without a biblical quote. Within a medieval, Christian context this is fine, but if you're looking at it from a modern point of view or say trying to convert Jews or Muslims, it's not very impressive.

That's why I like Anselm because he tries pure logic. There's no appeal to the Bible once in any of his works as he tries to prove the existence of God. Traditionally there are three proofs for God. The first is teleological, the modern variation of which is intelligent design in the United States: It's impossible for something as complex as nature to be created without some sort of creator in the background doing the work. The second is the cosmological, or in philosophy 101 is often referred to as "the unmoved mover." Something can always be traced back to something else: I was born from my mother, who is human, and humans originated from primates in Africa, which had mammals stomping on it for a while, which came from some earlier animal, which came from one-celled organism, which came from the Earth, which was formed from dust clouds coming together via gravity, which was made from stars exploding, which came from materials from the Big Bang, which came from... what? There has to be an end to this, someone who started all the movement, and that must be God.

Then there's ontological, which I'll admit is not very easy to understand. I wrote a fifty-page term paper on this fucker and I still have difficulties describing it, but this website has a good breakdown of Anselm's proof:

1. It is a conceptual truth (or, so to speak, true by definition) that God is a being than which none greater can be imagined (that is, the greatest possible being that can be imagined).
2. God exists as an idea in the mind.
3. A being that exists as an idea in the mind and in reality is, other things being equal, greater than a being that exists only as an idea in the mind.
4. Thus, if God exists only as an idea in the mind, then we can imagine something that is greater than God (that is, a greatest possible being that does exist).
5. But we cannot imagine something that is greater than God (for it is a contradiction to suppose that we can imagine a being greater than the greatest possible being that can be imagined.)
6. Therefore, God exists.

You can make arguments against that (a contemporary of Anselm's named Gaunilo of Marmoutier did, and Anselm wrote a long riposte), but I still appreciate the effort. He was able to step back and say, "Okay, let's say for a moment we come upon an atheist. Shaking the Bible in his face isn't going to change his mind because he doesn't believe in it to begin with. We need something that you can actually sit down and discuss on his terms. Let's a priori this shit up." Admittedly this isn't going to help when you're debating a Jew or Muslim, but at least it's a start.