There are many, many places to stay, including hotels, timeshare condos, resorts, and small motels. Your choice would be dictated by where you want to stay (which part of the island), your price range, and how much 'luxury' you want.
In Lihue, the Kauai Beach Resort is a nice hotel located about 5 minutes north of the airport. It is located near the shoreline, and access is from the highway via a private driveway, so there are no other developments nearby (other than condos).
Directly adjacent to the airport is a resort/ timeshare hotel, that used to be the Hyatt, then the Marriott. It is quite nice, and very centrally located.
You can also find lodging in Princeville (far north side of the island), Kapaa (10 minutes north of the airport), and Poipu (20-30 minutes south of the airport). Princeville is a planned community town with lots of timeshare condos and the Princeville Resort Hotel (Sheraton). It is a fairly long drive, but very pretty. Kapaa is a nice compromise, with several medium to cheap hotels and timeshare condos in the 'Coconut Plantation Marketplace' area. Yes, there are lots of coconut trees there. There is the Kauai Coconut Beach Hotel, which was a Sheraton, but now I'm not sure who runs it. There are several other smaller hotels in the immediate area, but I haven't stayed there so I can't attest to them. Between Kapaa and Lihue, the Kauai Resort is located just south of the Wailua River. It's an older hotel, but affordable.
Poipu is a resort area on the south shore part of the island. There are a couple of large hotels, lots of condos and timeshares, and resort type activities (tennis, golf, etc.) The range of prices fall from high to moderate for that area.
If you want to get away, Waimea Plantation Cottages are restored plantation houses. I've heard good things about this place, but I haven't stayed there. It's located in Waimea, on the south-west side of the island.
For a real adventure, try to get reservations at Koke'e Lodge (pronounced 'coe - KAY - eh'). This facility is a series of cabins managed by a company. It's located high in the mountains, where the air is thin and the cabins have fireplaces.
Unique Quality: Each has their own charm. Princeville is a resort town, populated by people who moved to Kauai from the mainland and other parts of the world. Not a hotbed of local culture. Poipu is also a resort area, so all efforts are towards catering to the clientele. Anytime you stay in a hotel, you're not going to get much authentic local culture. You need to get out and meet people and see the sights.
Directions: If you think of the island as the face of a clock, Princeville is at 12 o'clock, Kapaa is at 2 o'clock, Lihue is at 3 o'clock, and Poipu is at 5 o'clock. Waimea is at about 6 o'clock. Koke'e is locate