In General...
All magnetic storage works similarly.  The write heads align fields of magnetic particles on the surface.  The polarity of the field of particles will depend on whether a one or zero is being recorded.  Then when the read head passes over that area, it registers a one or zero depending on the polarity of the area.

Tapes
One of the first types of magnetic storage was that of the tape.  This uses magnetically coated ribbon, wrapped around two reels. There are two common forms of Magnetic Tape.  Either reel-to-reel or cartridge.  The cartridge type is pictured here, and usually has several hundred feet of magnetic tape, and is most often used in personal computers.  A reel-to-reel system is much larger, maybe several thousand feet of tape, and is on large spools.  They are usually used on large mainframes.  You may recall seeing pictures of refrigerator sized computers with silver and black reels on the front, with a black ribbon extending between the two reels..  Those are tape drives, albeit large ones.

A tape drive is basically like a cassette tape player/recorder.  It has two heads: one read, one write.  As the tape passes the head, the data is read or written, as the case may be.

Tapes have the advantage of being inexpensive, and rather reliable.  But they are slow, because they are sequential access devices.  This means you must advance or rewind the tape to the position where the requested data starts, much as you must advance and rewind a cassette tape to locate a song.  Tapes are also slow due to the nature of the media.  While data transfer can be increased by increasing the speed of the tape, this can lead to two problems.  One, the heads are more likely to miswrite due to not having enough time to align the polarities of the magnetic particles, or may misread due to the tape not being under the head long enough.  The other problem has to due with the durability of the tape media itself: The faster the tape is started, pulled, and stopped, the more it will stretch.  If the tape stretches too much, then it will render it unusable for data storage, and data loss may result.

Hard Drives

The first hard drive came about in 1956. It was invented by IBM (who else?) and could store an amazing (for its time) 5 million characters.  That is about five megabytes.  To put that in perspective, the average new computer today comes with a 20 gigabyte hard drive.  One megabyte is about 1/1000 of a gigabyte.  IBM called their machine RAMAC, or Random Access Method of Accounting and Control.  Random access was important: no longer did you have to wait until the tape was positioned correctly, a process that could take several minutes, but instead, you could access the data immediately regardless of where it was located on the hard drive.  RAMAC used 24 inch aluminum platters. Fifty of them.  Each platter was coated with magnetic iron oxide (oddly enough, derived from the primer used to paint the San Francisco Golden Gate Bridge).  This machine weighed almost a ton, and was the size of two refrigerators.  Assuming the machine used both sides of the platters, this would give a data density of about 90,000 bits/in².  Compare this to current hard drives which have densities of 30 million bits/in² or more.  Hard dives have obtained their smaller size simply through miniaturization.  Notes Tom H. Porter, chief technology officer at California based Seagate Technology's Minneapolis office, Smaller heads, thinner disks, smaller fly heights [the distance between head and platter]: everything has been about scaling," (Scientific American).

How a Hard Drive Works
 

How A Hard Drive Works

As can be seen from the figure, the heads of the hard drive move across the disk, being actuated by the arm to which they are attached.

A hard drive will often have an "access time" listed in it's specifications.  This is the time between first command and when the data can finally be sent back to the computer.  Access time is broken down further into "seek time," and "settling time."  Seek time is how long it takes for the heads to arrive in the general vicinity of the requested data.  The settling time is how long it takes for the heads to be still enough to read (or write) the data under the head.  To minimize settling time, it requires careful computation of the momentum the head/weight system will posses, as well as it's angular velocity, how much power will be required to bring it to a stop, and when the application of that power should start.  The hard drive could start to slow its heads when they are right over the data, but that would over shoot the location.  The application of torque in the opposite direction of motion must begin as the heads approach the location, so it can bring itself to a clean stop right above the data.  The better these computations are, the better the settling time is, and the better the access time.

Sometimes, portable storage of large amounts of data is needed.  Which brings us to optical storage...