What DR said. But also look for these common leaks:
Camshaft position sensor- It was originally white in color but a revised seal came out that is green and beaded. This is located on the driver's side of the head and has a 3 wire connector going to it. Item 15 in this thread:
This one will bite many people because when it leaks it will run down the head until it reaches the headgasket line. Then it will run to the back side of the block and down. It will look just like a headgasket leak but is a $15.00 repair instead of a $500.00 one.
Skill level 1
Valve cover gasket and spark plug tube seals- The spark plug tube seals will ONLY leak oil in to the tubes where your spark plugs are at, no external leakage will be found. Your valve cover gasket will leak pretty much on the back side. Most people don't know how to properly install this. They figure pull off old, install new and reassemble. NO! You must put RTV at 4 different points where the end cam caps meet the head in order for this gasket to work right. Skill level 2
Rear main oil seal- Will look alot like a headgasket because it will cover the bottom side of the block on the driver's side but if you properly clean the engine and remove the bell-housing inspection plate you should find heavy amounts of oil residue inside. Expect to replace your clutch too since it's probably covered in oil. Skill level 4
Oil pressure sending unit- There's a trend here... all your oil leaks are on the driver's side rear of the block. Don't get them confused. This is easy to fix but you will need an extremely large and deep socket OR must be willing to take a large wrench and grind off half of the handle. Skill level 1
Headgasket- Very likely suspect. The eclipses and talons never got the multi-layer steel headgasket from the factory and the one they did use sucked. Expect this to fail on you at 60K or more. Late built eclipses (98-99) may be able to make it to about 90K but it WILL fail on you. Skill level 6
Skill levels explained:
1- Newbie, very easy, minimal tools
2- Still easy, must be able to follow instructions and have a handful of tools.
3- Not for beginners. Modest amount of tools needed and experience needed.
4- Expect to spend half a day or more, wide variety of tools needed and access to service manuals or equivalent.
5 or more- You really regret having tried this one yourself. Some dealer only tools needed, you don't know if it will ever run again, HELP!!!