Oh boy... That's quite the question... Well, I'll start with my two characters' current ships, then I'll go in depth with previous ships.
My Federation character, Anplor, has a Reconnaissance Science vessel called the U.S.S. Lodestone. Because his species' society revolves so heavily around magnetism, it made sense to name it after a magnetic stone.
My Klingon character, Rameod, has a Qin Heavy Raptor named the I.K.S. Cha'zak, which is Hebrew for "strength".
Why Hebrew, you may ask? Well, I'm Messianic Jewish (which, to not go off-topic into too much detail, means I'm a Christian-Jewish hybrid). To that end, I know a decent amount of Hebrew; a language which bears slight similarity (in the way it sounds, anyways) to the Klingon language. All of Rameod's ships have Hebrew names with apostrophes and some random capital and lowercase letters to give it a Klingon feel. Some of Anplor's ships have Hebrew names, too, but aren't as coarse.
Now, on to the ship progressions...
*U.S.S. Northstar, Light Cruiser: Since Anplor hails from Polaris, I thought the name fit well.
*U.S.S. Tengam, Science Vessel: The name of Anplor's homeworld.
*U.S.S. Rofey, Research Science Vessel: Hebrew for "doctor".
*U.S.S. Shalom, Support Cruiser: Hebrew for "peace".
*U.S.S. Sojourner, Long Range Science Vessel: An adventurous name for a ship famous for adventures.
*U.S.S. Lodestone, Reconnaissance Science Vessel: (See above)
My main ship is the I.S.S. Conquest, a name I picked to reflect the origin of the ship and crew as from the Mirror Universe. I wanted a name that would not be selected by the Federation, but at the same time something the Terran Empire would find admirable, and the Federation wouldn't reject completely (and Conqueror was both a bit too common, and something that sounded too personal for a ship). Also, I liked the idea of a TV show named "Star Trek: Conquest" to give a different feel.
The ship prior to that I named the U.S.S. Dawnstar after one of my favorite characters in graphic fiction, as well as a fitting name for a ship of a class from the dawning of the Transwarp era.
Another ship, the U.S.S. Bullet is a homage to one of the most heroic space battleships in all Sci Fi novels I have read, W. Michael Gear's Spider trilogy of star-spanning war and spirituality (not for young readers...the author doesn't pull -any- punches either with war or the ways of evil).
Finally, my Galaxy-class ship is named the U.S.S. Legacy, after a ship that was to be the focal point in an old Atari 2600 game that was never released, but that fired my imagination as a child. I'm still waiting for Final Legacy to finish beta-testing.
My Fed's name is Gordon Rodgers, (spelling deliberate) which if you understand the homage, it should be all that's needed to explain his ships: the USS Raymond Gunn, the USS Ray Gunn, and the USS Buster Crabbe.
My Orion flies the IKS Black Horizon, which came to me as I was staring at the space background in one of the corners of the galaxy. I considered looking up the Klingon for it, but opted not to.