Both leading ridesharing services Uber and Lyft are operating on the island of Oahu (Honolulu). Uber is also available on Maui, but Lyft is not available on Maui as of this writing. Neither ridesharing service is available on Kauai or Hawaii Island (the Big Island) yet.
Note that Uber and Lyft rideshare drivers are not allowed to pick up passengers from Honolulu International Airport. Only sanctioned taxi companies and their drivers can pick up people at the airport. Although there are popular hacks to circumvent this restriction (taking rental car shuttles off airport property, walking to nearby businesses, etc.), periodic stings by law enforcement make most rideshare drivers reluctant to accept airport requests. On the other hand, all Uber and Lyft drivers can drop off passengers at the airport.
In addition, the Honolulu City Council, like other municipal governments, has proposed ridesharing regulations that Uber and Lyft have said would threaten their ability to operate. So far, however, these regulations have not been implemented and both companies continue to operate.
Note that Uber and Lyft coverage is concentrated in urban areas, although there are usually cars available in larger neighborhoods in Leeward, Windward, and East Oahu. Drivers will usually respond to requests from even more rural locations, but you may find yourself waiting 20 minutes or more.