A couple of things to check-

The most common and easiest is a stiff link that will feel like skipping. Especially after removing and replacing a chain- did you relieve the pin after driving it back in? Check that all the links in the chain are free-moving. It there's a stiff one, then drive the pin back through the plate a tiny bit to get it to swivel freely.
The next most common is worn cogs and chain. This will be worse when really mashing on the pedals, and worse in one cog than the others. Do the cogs look symmetrical or have they developed shark-fin looking teeth? If they look worn, then they are worn. The fix is to replace the chain and cogs (both because a new chain alone will wear out instantly when running on worn cogs, and skip even worse). Check the chain wear- hold a ruler against the pins and if the chain is more than 1/8" in 12" elongated, then it's well worn out. I replace my chains at 3/32 inch wear and go through about 4 chains per set of cogs in a 9-speed bike.