The leader is the length of nylon or fluro, usually tapered, that makes the "invisible" connection between the fly and the tip of the flyline. The tippet is the thinest part of the leader the the fly connects to. I generally start with a store-bought tapered leader of 9 or 12ft and chop the last few feet off and finish the end in a tiny loop. I can then add a couple of feet of whatever tippet I'm using with a loop-to-loop and replace it as needed. As you keep changing flies, the tippet will get shorter.
You can also setup your own tapered leaders using lengths of different diameter mono. Get hold of a copy of "Practical Fishing Knots" by Lefty Kreh and Mark Sosin. It's a real gem and will explain all you need to know in easy to follow diagrams.
I've never used eyelets and never met anyone who has. I wouldn't trust them personally but that's just my opinion.
When sight fishing, look over your shoulder from time to time, you never know who's behind you