Dips: A dip is a sudden down-and-up, such as a wash or creek. The key to smooth dip-riding is the position of your body relative to the bike.
As the bike drops over the edge of the dip, keep your body vertical. Let the front end of the bike passively fall away from you. Let off the brakes. Your weight is on your feet. Feel the saddle press forward a bit between your thighs, but don't go into the full "downhill steep" position. You and the bike are in free-fall.
During the drop, let your body rise up, so you're standing with the pedals level when you reach the bottom of the dip. As the bike flies into the bottom of the dip, let it rotate under you. As you feel the uphill slope, transfer some weight to your hands, and let your body come forward.
Let the bike rise up under you as you absorb the uphill hit. The harder the bike bucks up against you, the more weight you transfer to the handlebars.
As your front tire reaches the top at the far side of the dip, put as much weight as you can on the handlebars. (You don't want to wheelie here!) If you don't have quite enough momentum to carry you over the top, start pedaling. Once the rear wheel has climbed out of the dip, let your weight settle back again.
Dip with a Tire Trap: What if the dip has a deep washout at the bottom? If your front tire drops right to the bottom, you endo. To safely ride through this type of dip, you need to "throw" your front tire across the bottom.
As the bike drops over the edge of the dip, let the front end of the bike passively fall away from you. Keep your body back a tiny bit further than you would with an ordinary dip, but (most important) let your body rise up, so you have room to work the bike underneath you.
When the front wheel is a couple of feet from the bottom, pull up hard on the handlebars. You're pulling the bike up, but also back a little towards you. The objective is to get the front wheel off the slope, so it flies directly across to the opposite side of the dip.
Because you pulled up and back on the handlebars, your body moves forward relative to the bike as it travels across the bottom of the dip. Good. That positions your weight so the rear wheel can also clear the trap. (If you fall back onto the seat -- as with a "wheelie" -- your rear tire will slam hard.)
As the front tire has touches the opposite slope, your body is forward, and it "pulls" the rear of the bike passively over the trap. Once the rear wheel is on the uphill slope, transfer more weight to your hands. As the rear wheel rises over the far slope of the dip, let your weight settle back again.
The Bunny Hop: The bunny hop is a way of clearing small obstacles, or holes. See the section on tire traps. A bunny hop is done by throwing your weight down on the pedals to "jump" your body up into the air, then dragging the bike up under you by lifting up (gently) with both your feet and the handlebars.
Sprint to increase your speed. Keep your eyes high, looking past the obstacle. Get up off the seat, and level your feet on the pedals. (Most of us have a "preferred foot forward" -- if we usually jump with the right foot in front, it seems clumsy having the left foot in front. Practice bunny-hopping while alternating your leading foot.) Center your body over the crank, feeling the weight in your feet. About a foot before the front wheel reaches the obstacle, jump as if you're going up for a basketball rebound. The bike will compress down (with the weight going straight through the crank), and both wheels will rebound up equally. Lift up lightly with both hands and feet, then wait for the bike to return to earth.
Bunny Hop with a Twist -- The Side Hop: When riding over a root at a sharp angle, you may slide one of the tires out. You can clear the root with a one-two (see Dolphin Hop, below), or if you have enough speed, you can side-hop it. The front tire will go straight over the root, while the rear tire will move to the side, dropping down as it crosses the root sideways.
The Dolphin Hop: The dolphin hop is a variation on the bunny hop, designed to carry you over an obstacle that's a bit too high to clear with the wheels level. The dolphin hop is usually done as a "one-two" sequence -- the front tire jumps over, then as the front tire lands, the rear tire jumps over. Expert jumpers can do the whole thing in the air -- the bike starts the jump aiming upward (so the front tire clears), then completes the jump with the bike aiming downward (so the rear tire clears).
Get your speed up. Get your fingers OFF the brakes. Just before you reach the obstacle, compress the pedals with both feet. As the bike rebounds, pull up on the handlebars so the front tire raises up, higher than the obstacle. The higher you can yank the bike's front end, the better this jump works.
Right as the front wheel crosses the highest point of the obstacle, pull your hands a bit back towards you. (This seems counter-intuitive, but try it.) Let the front wheel fall down. Let your feet drift up a tad, but they must stay in contact with the pedals. The rear wheel will rotate up as the bike pivots under you -- with both wheels in the air. The rear tire will rise up, clearing the obstacle.
As the bike lands, keep the front wheel absolutely straight and don't touch your brake. The bike will land front-wheel first. A tiny tweak of the front brake, or a slightly turned handlebar, can make the front wheel dig in. Endo.
See the section on Tire Traps for more information about using the bunny hop and dolphin hop.
The Crank Grind: Some large logs require the use of the large chainring as a "third wheel" to get across. This maneuver is best reserved for strong bikers with good balance and enough money to buy new chain rings.
Approach the log with a mighty wheelie. Touch the front wheel down on the backside of the log -- the wheel should hit the far side of the log, not fly across it. Throw your weight onto the handlebars and raise your body up. Once your weight is above the log, grind the crank over the log. As the rear tire contacts the log, "throw" the front end forward, letting the bike rotate under you. Try to set the front tire down as far from the log as possible.