Most popular synthesizers use "subtractive" synthesis, meaning their oscillators generate a harmonically rich starting point, such as a square or saw wave - think MiniMoog. This sound is then shaped and sculpted using the instruments filter section, where you carve away at the sound using the filter cut-off and resonance to remove or accentuate existing frequencies. FM works in an opposite fashion by building up harmonically rich sounds from a starting point without any complex frequency content. FM usually begins with a sine wave oscillator, in FM this sinusoidal starting point is called the "carrier" and contains no harmonic complexity.
From here, another oscillator is used to modulate or rapidly move the frequency of the carrier (think of it as extremely fast vibrato). How fast this "modulator" oscillates is controlled by its "ratio". If you slow the ratio right down the carrier will slowly pitch up and down, similar to how an LFO would modulate pitch. However if you increase the ratio, the frequency of the carrier will osculate faster and faster and new frequencies (partials) begin to appear either side of the fundamental frequency, these are called sidebands. These sidebands give the output signal its harmonic content. The important thing to remember from all this is that you never actually hear the modulator; you only hear its effect on the carrier.
The next step is to create more complex sounds by changing the rate of these modulations over time using envelopes. Typically, each oscillator (modulator or carrier) is paired with its own envelope generator, which controls the oscillator's output amplitude. In FM this combination of elements is called an "Operator". The more Operators a synth has the more complex sounds you can create.
As mentioned earlier the higher the number of operators available, the more synthesis options you have. This is achieved by configuring the operators in different routing arrangements (what modulates what). For instance you may have a four operator synth (operators A, B, C & D) and configure operators B, C & D to all modulate the carrier A. You may also have a configuration where A & B are carriers and C is modulating only A while B is being modulated by both C & D. These configurations of operators are known as "algorithms". Most FM synthesizers have a set number of selectable algorithms; the more operators available, the more routing options you have within an algorithm and the more complex your patches can become. For example Digitone has 8 selectable algorithms, Ableton's Operator has 11 while the classic Yamaha DX7 had 32, however some synths do actually allow you to create your own algorithm configurations.