It’s one of the simplest beats to learn on drums but a bit more challenging to execute with finesse. Simply take
a 2/4 bar and play 16th notes on the snare drum hand to hand (RLRL), counting "1 e and ah, 2 e and ah".
Then put the bass drum on beats 1 and 2, and accent the snare beats on the "and" of 1 and the "and" of 2.
(The following exercise also shows an alternative accent on the "ah" of 2.)

Many drummers play this beat with wire brushes, plastic brushes (Blasticks), or Promark’s Hot Rod’s on snare drum.
Also, many play the train beat with a slight "edge" to it (just short of rushing it a bit). This gives it a forward
momentum and will sometimes drive the tune a bit better.

You can play the train beat with doubles, paradiddles, or all with the right hand while the left plays the backbeat. You can also add a "bounce" to it and essentially play it with swing eighths. (Common in rockabilly tunes) This will then become a basic shuffle played on the snare drum.

The Devil went down to Georgia – Charlie Daniels
Country Boy - Ricky Skaggs
On the road again – Willie Nelson
Petticoat Junction – TV show
Last train home – Pat Metheny
Two Step – Dave Mathews Band
Gallows Pole -Led Zeppelin
Train Time - Cream
Trouble on the Line - Sawyer Brown
Train Kept a' Rolling - Aerosmith

Resources: 

Video: Country Drumming (Volume I) Charles Eaton (93-DJR-60) $29.95
This video covers the basics of country drumming. Charles demonstrates how to do the
" train beat", the country shuffle, two beat, etc. on drums.