good questions those. Fusion is a matter of taking two particles, and moving them towards eachother with a high enough energy to overcome the natural repulsion (electromagetic forces between the protons) to be bound by the strong nuclear force (attraction between protons/neutrons). So you can have a single pair of atomic nuclei fuse if you smash them together just right. Doing this in a particle accelerator you can have very low densities, and relatively low energy (just enough to fuse).
That's it. The other factor comes into play is how often you want a fusion event. If you want the rate of fusion events to increase, you have to have more atoms. If you cram them in a smaller space the atoms will have to travel less distance before they get another chance to smack together. This is where the pressure comes in.
If you increase the pressure you increase the number of chances an atom has to fuse. Every failed attempt changes the energy of the atom, sometimes increasing it, sometimes decreasing it, as the momentum is swapped between it and the other atoms it hits. Eventually it'll get enough energy that the next atom it hits will be with enough energy to fuse. Increasing pressure increases this probability.
The other way you can increase the probability of a fusion event is to boost the temperature. Temperature is a measure of the average kinetic energy of the atoms being observed. Increase it, and the atoms move faster, meaning it's more likely for them to smack and fuse than smack and bounce.
Gravity kicks in because it acts like a lid. A ball of gas, heated up, expands and the pressure drops. Without some 'outside' force acting to compress the ball of gas it just spreads out and fusion can't occur. In the case of a star the gravity of the gas itself is enough to compress part of the gas (that in the core of the star) to keep the pressure up, like the lid in a steam cooker. More gravity means a higher pressure can be maintained, meaning a higher rate of fusion...
Get the probabilities high enough, via manipulating these two values, and you can get a sustained fusion reaction, where it generates enough heat on it's own to maintain the temperature required.
To a certain extent you can have a trade off between the two, though the temperature has to have a minimum value for fusion to work at all. This can produce various rates of fusion. Small stars, despite smaller reserves of hydrogen, burn for billions of years. Massive stars (like Sirius or Betelguese) have huge reserves of hydrogen, but their fusion furnaces burn so fiercely that they only last on the order of a hundred million years!