Awesome question! This is one of the most misunderstood concepts in all of weight lifting. I'll be glad to give you some insight.
Forearm strength is all about GRIP! Also, there are two muscles involved, your "forearm" is the muscle underneath, and the "forecep" is the muscle on top. Your forearm muscles control your hand (grip) strength, as well as your wrist flexion (forearm) and extension (forecep).
Traditionally, many people do wrist curls with a dumbbell, which works just fine, but there are tons of other exercises that give you veined-up jacked looking forearms.
1) DO NOT lift heavy with lifting straps. If you deadlift or do pull ups, use just your grip. Do not use the straps. The straps take all of the work of your forearms away since they eliminate your grip strength from the equation. If you can deadlift 315+ without wrist straps, chances are your forearms are like rocks.
2) For my wrestlers and grapplers in general, there is a cheap grip method I have them do while they're at school or watching TV. Take a sheet of newspaper and crumble it in your hand. Keep squeezing as tight as you can for several minutes until you can close your whole hand around the crumbled ball. Your forearms will be on fire.
3) On all weight lifting exercises, make sure your hand has a firm grip on the bar or dumbbell. If you are hooking your fingers around the bar lazily and not squeezing the bar with your whole hand, you are letting a lot of forearm work fall through the cracks.
4) Next time you do pull-ups, wrap a towel around the top of the bar and try to do pull ups from the hanging towel sides. This makes it MUCH harder to grip, and again, your forearms and foreceps will be burning like you wouldn't believe.
5) Take up rock climbing. I trained a 16 year old who rock climbed and the kid had a grip like a robot. He literally almost crushed my hand, and he wasn't even very muscular. His forearms on the otherhand, looked like they were straight off the Hulk.
Give these tips a shot. I hope this helps bud!