Class A refers to the operation of the output devices and is neither a tube or transistor thing- its both.
Class A makes the most heat and the least amount of power, but is also the lowest distortion. Class A amplifiers can be either single-ended or push-pull, transistor or tube, and can or may not have an output transformer.
A class A transistor amp does have a better chance of sounding like a tube amp as it will be lower distortion. A lot depends on design though- to sound like tubes you have to reduce odd-ordered harmonics, something that is challenging in the world of transistors, but not impossible.
Class A does tend to have more authority from my own listening. Ultimately I recommend that you use your ears rather than anything that you read or the like. Class A is the lowest distortion, but will be the best sounding only in concert with a lot of other design features.