A transformer is a passive device which can change the voltage of an AC current which is fed into it; the output can be either higher or lower voltage, depending on the details. (Of course, conservation of energy dictates that if the output is higher voltage, it will be a smaller current, and vice versa.)
Transformers also isolate the input and output circuits; i.e. the input voltage is measured across the two input leads; and similarly the output across the two output leads. However, unless there is some other connection between the two sides, neither lead on either side has any voltage relative to either on the other.
(It is possible to have a transformer which has the same output voltage as the input; these are called 'isolation' transformers, since it is the isolation property of a transformer which is being sought in this case.)
A transformer consists of two inductors ('coils') wrapped around a 'core' - an element through which magnetic fields can easily pass. (The core is usually a loop, so that the field in it circles around and joins itself.) When a current flows into the first coil, it creates a magnetic field, which flows around the core, and induces a current in the second coil.
If the output coil has more loops than the input, the output voltage will be higher, and vice versa.
It is common to put a tap in the middle of the second coil (called, logically enough, a 'center tap'); the voltage at one end of that coil, at any point in time, with reference to the center tap, is the exact opposite that of the other end; i.e. the waveform at one end is the exact opposite of that at the other. This is useful when creating DC using a pair of diodes (one from each end).
Similarly, for transformers which take as input 'wall power', the input is sometime wired as two separate coils. When connected in series, it can take 220V AC as input, and if they are wired in parallel, 110V. This allows easy conversion from one power type to another.