There have been some very thorough posts here, but I'm gonna take a swing at this and see if I can simplify things a bit...
Chordwise Flow
Remember in basic aerodynamics; the curve on the top of the wing accelerates the air? Well that's great and is the whole point for the shape of the wing, but when you get going too fast it can cause problems.
On a straight wing the movement of the air, for the most part, is from the front of the wing to the back of the wing. This is called chord wise flow, because it is parallel to the chord of the wing.
Spanwise Flow
What a swept wing does is trick the air into thinking it is also moving along the length of the wing, from the root towards the tip. This is called spanwise flow because it is parallel to the span of the wing.
When the air moves from the root to the tip it doesn't encounter any curvature in the wing (camber) and so it doesn't accelerate. This allows aircraft to fly faster without having the air above the wing reach the speed of sound.
Critical Mach Number
When any air around the aircraft reaches the speed of sound a shockwave is formed. The speed at which this happens is called the "Critical Mach Number." Most pilots confuse this term with a lesser known term: "Force Divergence Mach Number."
The force divergence mach number is the speed at which the shockwave is strong enough to cause some real problems with drag and controllability. This is the speed that most people incorrectly refer to as "Critical Mach Number." Don't lose too much sleep over this one 'cause most people don't know it anyway.
Bow Waves -vs- Normal Waves
Unless you fly a supersonic aircraft, the only shockwaves that you will be concerned with are the ones on the surfaces of the wings. These are called normal shockwaves because they are at right angles to the wings' surfaces.
The type of shockwave that forms on the nose of the aircraft (or the front of the wings) only happens when the whole aircraft reaches the speed of sound. These are called bow waves. (To be more correct, this type of pressure wave is actually ahead of you everytime you fly, even in a Cessna, the only difference is that supersonic jets can actually catch up to it and make it stronger creating a shockwave.)
In reality, you don't worry about all of this on a daily basis when you fly a jet. You just make sure you don't fly too fast or too slow or too high for your weight and temperature. If you're in an advanced jet, the computers figure this all out for you. If you're flying an older airplane, you have to look at some charts once in a while.
Swept Wing Problems
There are really only a few downsides to sweeping the wings. One is a tendency to dutch roll, which is usually taken care of by a yaw damper, which is basically an autopilot for the rudder that keeps the plane from yawing around and making everyone sick.
Another is that swept wings don't provide as much lift at slower speeds so the flaps tend to be more elaborate and most jets also have "slats" that are sort of like flaps on the front of the wings. They both help to create lift at slower speeds so the jets can take off and land in a reasonable distance.
One more is that airplanes with swept wings have really bad stall characteristics. Because the tips tend to stall first, the center of lift moves forward and the airplane actually pitches up; just the opposite of what you want. Most swept wing airplanes have "stick shakers" that cause the control yoke to vibrate, simulating a stall buffet so that you can recover early. Some airplanes even have "stick pushers" for pilots who don't clue in quickly enough. A pusher will actually push the yoke forward for you so that you can't enter an unrecoverable deep stall. (I won't even get into Airbus fly-by-wire control laws.)
About as clear as mud? Ah, just keep the shiny side up and everyone's happy. Hope this helped.
