Yes, 2A3 is a type of tube. It puts out 2-3W of power - enough to drive a pair of efficient speakers to ear-bleed levels, let alone headphones.
The classic tube sound is indeed warm, with slightly rolled off highs (which isnt the same thing as NO highs, mind you) & luscious, rich full-bodied mids. The performance in the mid-range is truly magical and absolutely makes the music come alive in a way that no solid-stage amp can match. Tube amps dont measure as well and have a lot of 2nd order distortion - which feedback circuitry in solid-state designs remove. A lot of people, myself included, feel that this 2nd order distortion is what restores a sense of body to the sound and makes it match more closely to what one hears in a concert-hall, even if it doesnt measure as well.
However, there are a lot of misconceptions about tubes - the moment people see a tube, they think it is going to have the above sound. That is not true.
You can get push-pull tube amplifier designs which measure and perform very similar to solid-stage designs - the old Dynaco ST-70s were a good example of this, as are most modern high-power (35W+) tube amp designs.
The classic tube sound mostly applies to single-ended tubes, which are essentially have one driver tube and one single gain stage, with zero feedback circuitry.
The crazy thing is, for whatever reason, single-ended tube amps are stupidly expensive bought off the shelf (the cheapest I have seen is the Decware Taboo). I built my own using around $500 worth of parts, and it was comparable to a commercial amp costing around $5000 (I know cos I opened up a couple and checked out the circuitry inside). So if you are on a budget, you might want to pick a design and either DIY it or buy something like the Bottlehead kit.
Now, whether or not you like this depends on what your baseline is. Mine is live classical concerts - I want a sound that replicates that, as opposed to reproducing exactly what is on the CD. If you are used to electronic or amplified music, where there is no objective reference (other than what the sound engineer decides on the CD), that's a different story.
Hope that helps.