If you want to dive right near your hotel, stay at the Ambassador. They're a 4-star hotel. The Red Sea Sports Club is on-site. Some of the diving is right accross the street. The Ambassador is a nice family resort. The location is on Coral Beach.
I'd recommend spending a day at the dolphin reef. In addition to seeing the dolphins, you can take a dip in their Dead Sea-style pools. It's hard to explain, but really worth it.
The best resort in Eilat is the Sheraton Herod's Palace Hotel. It's got all the usual 5-star luxury stuff, including a great spa. Be aware that it is expensive. Herod's is located on North Beach, further away from the diving, but right in the middle of all the hotels and restaurants. By comparison, the Ambassador is more isolated and is best if you have a car.
There are other nice hotels like the Dan, Crown Plaza, King Solomon's Palace. They are all on North beach. A taxi ride to Coral Beach is only 5-10 minutes and costs 20-25 shekels.
My favorite diving is in Taba, right over the Egyptian border. Aqua-Sport has dive centers in Eilat and Taba and can take you over. It's very easy. In Eilat, they are on the beach, accross the street from the Ambassador/Red Sea Sports Club.