A magnetic strip is a thin layer of magnetizable material which is placed on credit cards and driver's licenses and bank ATM cards and train tickets and hotel room keys and lots of other places. I've even seen paper telephone cards in Italy which use a magnetic strip on the back.
The magnetic strip is a lot like a tiny bit of magnetic recording tape, like that found in a music cassette or a videotape, except it is glued to the card. It is a plastic material with magentizable powder mixed in with it. The strip can be magnetized by exposing it to a strong magnetic field. Bits of it can be magnetized in different directions by a device which generates a strong magnetic field over a short distance. Read/write heads of a tape recorder do exactly this -- they have coils wrapped around special metals to make an electromagnet. Passing a current through these coils generates a magnetic field which can magnetize bits of the strip. The strip can have many many little North and South poles on it, and their numbers and locations are used to encode information (such as your credit card number or bank account number or driver's license number). For security reasons, your passwords or account balances should not be recorded on the strip because just about anyone with proper equipment can read the strips.
The strip is read with a similar (or even the same) device which wrote it. The process works in reverse -- by moving the strip back and forth over the read head, voltages are induced in the coils (Faraday's law describes how this works), and these can be amplified and recorded electronically, so that the information can be sent to a computer.
Sometimes magnetic strips do not work all that well. If you put them next to a magnet, they can get erased. Don't carry refrigerator magnets in your pocket next to credit cards and drivers' licenses! If the card gets bent or wrinkled they can be hard to read (the strip pretty much has to touch the read head of the reader, and if the card is bent it may not touch everywhere on the strip). Heating the card may make the strip lose its information, but most plastic cards will melt or deform first, causing problems. Don't leave them out in the sun or on a car's dashboard in the hot sun. They can also fail to work well when they are worn down from many uses.