Having gotten into the Famicom quite a bit lately, I had to start this topic.
Gradius II - One of the best 8-bit shmups ever, pulling off graphics and sound unheard of on an 8-bit machine. The control is precise as well.
Crisis Force - The other greatest 8-bit shmup. It pulls off even more impressive graphics than Gradius II, and controls just as smoothly.
Parodius Da! - I didn't think that the FC could pull off a decent version of Parodius, but I was wrong. Too bad you can't deactivate the damned Roulette feature, though.
Getsu Fuu Ma Den - A fantastic action/platformer in the vein of Zelda II, but a lot darker than most FC games. Knowledge of Japanese is essential to win, though.
Waiwai World 2: S.O.S.!! Parsley Castle - I loved the idea behind the original Konami Waiwai World, but the execution wasn't so great. #2 more than makes up for that, though, with the ability to select the characters you want to take with you (some of Konami's most famous heroes - among them Simon Belmont, Bill from Contra and Goemon), and better control.
Bio Miracle Bokutte Upa - AKA "Baby Land". It's a platformer that stars Upa the baby, and has some fairly unique gameplay for a platformer.
Splatterhouse: Wanpaku Grafitti - Aside from being a great action game, WG is hilarious. Movie parodies are everywhere!
Yes, I realize that almost every game I listed was by Konami, but that's just because when it came to the Famicom, they were the king.