I decided to take the time to explain exactly what a synthesizer is and how it works. I hear far too often that it is just a computer and that it is not a real instrument. I'm here to reject your reality for you and implant the truth.
A synthesizer is basically a sound producer. There are a few basic components to a synthesizer: Oscillators, Amps, Envelopes, Filters, Mixers, and usually effects.
Oscillator - an oscillator (or osc) is almost always the primary sound source for any synth. The output of an Oscillator can be something very simple, like a pure wave, which is pretty much any of the basic waveforms: Sine, Square, Saw, or Triangle. Oscillators can also be supplied a sample, such as what is used in sampling synths or samplers in general.
Amp - An amplifier simply amplifies a signal, or makes it louder. Once you've got a signal coming from your oscillator, you will probably need to amplify the signal in some way in order to make it audible. In analogue synths, amps convert a voltage into audible sound. In virtual or digital synths, amps can often just act as volume controls because the amps are often integrated into the oscillator.
Envelope - An envelope is simply a way to control your sound over time. This is usually called an EG, or Envelope Generator. Most likely you will also want to use an envelope in order to control the way in which the loudness of the signal changes over time, but you can also use envelopes to control how filters and even oscillators react. This will have an enormous effect on the sound, and is one of the most important factors in determining the overall sound. Let's go a bit more in depth with this one.
If you compare the sound of different instruments, one of the many distinguishing factors is the way in which the volume of the sound changes over time. For example, an orchestra of strings tends to kind of fade in, whereas a guitar reaches peak amplitude, or volume, almost instantly. Similarly, after a note is not being played (aka the bow is not bowing nor the pick picking), there is some time between when the note stops being played and when it isn't sounding, usually as a fade out, or gradual decrease in volume.
Envelopes can control such things as pitch or velocity (how hard, and thus loud, the note is). The effect of these is similar to keyboard tracking: you can have the envelope's character change as you play up the keyboard or play notes louder. Envelopes can also be used to control many other parts of a synth whose parameters you want to change over time. For example, you might want to use an envelope to control a filter, so that the filter opens up over time as you hold a note. Other uses might be controlling the pitch or pan values of a note over time. As you can see, Envelopes can be pretty awesome not only for reproduction of real instruments, but for making some neat sounds.
Speaking of Filters...
Filter - a filter removes certain parts of a signal, just as a sieve will remove certain parts of a fluid. You can control what gets filtered out of a liquid by choosing sieves with different sized and shaped holes, right? You can do the same depending on choice of filter, the settings used, and what controls the filter. The effect of filters can indeed be incredibly powerful. Some people feel that filters are the most important feature of a synthesizer, and with good reason: filters can totally change the tone, acting as everything from an Equaliser to a way to make your sine wave sound like a crying child (it has its uses...). What is super important to know is that filtering does not just have a 'reductive' effect.
Filtering has a much more dramatic effect on your sound than simply taking things out because, due to how some can change qualities of the harmonics within a sound, it can not only reduce some harmonic overtones but it can also boost others. This results in sound that can differ greatly from the original. Some standard filters are called Pass Filters, which act as EQ settings, allowing only certain parts of the sound to pass through it. Low pass filters let only low frequencies pass, high pass filter allow high frequencies through, and band pass filters allow whatever the specified frequency only to pass.
LFO - LFO stands for Low Frequency Oscillator. Like the Oscillators described above, a low frequency oscillator outputs an oscillating frequency of some sort, usually in the same waveforms as the normal oscillators can, but the output of an LFO is generally much slower than that of a standard Oscillator (hence the name, Low Frequency). When you hear the output of a Sine wave from a normal sine oscillator, the frequency of the wave is generally so fast that you can't distinguish the individual cycles of the wave, you just hear a continuous tone. With an LFO, the cycle of the wave is so slow that you can (at lower frequencies anyway) actually distinguish the individual cycles of the wave. LFO's are generally used to modify some aspect of another part of a synth: the pitch of an Oscillator, the amplitude or panning of a signal, the application of a filter, et cetera. For example, using what I just described, LFO's can create a tremolo effect, vibrato, or auto-pan effect.
The actual uses of an LFO are pretty much infinite, though. With older patching synths, you could literally attach the LFO to whatever you wanted. With modern virtual patching, you can connect one source, in this case our LFO, to many things -- thus, one LFO can now control every aspect of a synth at once.
Mixing - Just like mixing signals from a band in a studio with a standard mixer by adjusting volumes, most synths come with a mixer for at least the oscillators. Some synths even have mixers for effects and filters affecting the wet/dry-ness of the signal. What this refers to is the amount that part of the synth affects the sound. Wet means that the signal is totally affected by the effect or what have you, and Dry means that the signal is totally unaffected.
Level - talking volumes, I should probably also mention what level means. Level usually refers to volume or mix. Volume, obviously, refers to how loud something is. Mix is like that effect mixer I was talking about above.
Effects - effects simply change the sound, very much like filters. Some effects don't remove anything, but add to the sound. Standard effects, such as flanger, phaser, chorus and delay/loop, have this affect. There are some more powerful effects such as Downsamplers that remove parts of the sound. Let's talk about some of these standard effects:
Flangers are kind of like a slow LFO affecting the pitch of a note's overtones.
Phasers shift the oscillations of a sound, in a more extreme way that flangers.
Chorus effects are liek a fast version of flangers. It gives the effect of being slightly but still musically out of tune.
Delay takes a sound and replays it after a set time. Delays tend to lower in volume over time.
Loopers are very much like delays, but they never get quieter.
Downsamplers literally lower the bit rate, or how fast the sound renders. At extreme levels, this can create a sound similar to slowly scraping metal... or a pissed off computer.
There are more things that I could go over, such as MIDI implementation, how synths know what pitch to play, various modifiers, et cetera, but I won't. At least, not now. This is just an overview of the basics of synthesis. Something to note is that all the parts that make up a synth are naturally created by normal instruments. Synthesizers allow the near-infinite tweaking of these parts to create equally infinite sounds.
If you wish to get into synthesis, I recommend getting Gunnar Ekornas' Minimogue VA (http://home.no/gunnare/). It has several of the elements listed here in a very easy to understand layout. It was actually my first synth. Just don't get the Luxus version. I've had problems with that. When you download it, you may also need to get something called savihost (http://www.hermannseib.com/english/savihost.htm), which will allow you to use the synth as its own program -- no installation needed. The instructions on boths sites are pretty clear, but just in case:
1. download the minimogue
2. download savihost
3. put both the one folder
4. name savihost.exe MinimogueVA.exe, or whatever exactly the dll file is named.
5. run minimogueva.exe.
6. enjoy synth happiness. It comes preloaded with many sounds, so you can look at the settings and get a feel for how the controls affect things.