When you talk about the city of Honolulu, the county of Honolulu, and the island of Oahu - you're actually talking about the exact same land area. Honolulu - the city - is the capital of the entire state of Hawaii and has by far the largest concentration of its residents, with roughly 80% of Hawaii's population.
Deciding where to stay in Honolulu is essentially a question of choosing between Waikiki or somewhere else. Most of the hotels in Honolulu are in Waikiki, and you may never need to leave this small neighbourhood to find everything you want to do, see, eat, and buy during your holiday.
But, if you opt to stay further from Waikiki, you can still take advantage of all Honolulu has to offer, including the incredible white sand beaches (and all the water sports that go with them), a vibrant nightlife scene, the biggest open-air shopping mall in the country, and the military monuments at Pearl Harbor. Don't miss out on the array of cuisines available in Honolulu, either, from traditional restaurants to the farmer's markets to the city's famed shrimp trucks.
Most visitors to the Hawaiian islands enter through the airport in Honolulu - so even if the rest of your holiday is to be spent elsewhere, it only makes sense to stop in Honolulu for a few days before moving on.