"close harmony" simply means that the chord notes are all within the same octave(the alternative is called "open harmony")-"counterpoint" means that parts are moving in different directions, and when you have four voices sounding at once, it is pretty hard not to have some counterpoint going on--

For a number of reasons, in barbershop arrangements, most of the voices tend to move as little as possible from the place they started--they tend to stay on the same note til the chord change(unless they are the lead), and then they move to the nearest note in the next chord--

The typical song would be "Down By the Old Mill Stream", which starts off by sounding,one note at a time, the G-C-E inversion that JimmyT mentions above(down by the) then adds a G(old) then everyone moves down by half steps for "Mill" and "stream"(the bass stays on the G on the bottom throughout)--if you can do that, you've got the essence of barbershop--but ear fatigue sets in pretty quickly--


"Barbershop" is "novelty music"--which means that it is easy to recognize, gets it's applause with a few stock musical devices, and then is over--it evolved the vaudeville/variety Olio acts, that would sing quick, uptempo, familiar songs in front of the curtain while a set change was going on(or to seque out of an act that bombed)--like most novelty music, the gimmicks wear thin after a couple numbers--

Still, it is fun, and it's roots are in both in the classically based popular quartet music of the late 19th and early 20th century(which often featured many more than four voices) and in the multipart, social singing that used to take place, not so much in barbershops, but in saloons, before recorded music ended the practice--