Under what conditions does hydrogen begin to fuse?

It needs to have sufficient energy that the hydrogen atoms can overcome the very strong electrostatic attraction between their nuclei- this requires very high temperature.

It also requires very high pressure to initiate and sustain fusion, and usually the reaction must be contained to remain viable- as the hydrogen heats up, it will expand if uncontained, which will cool it down. In stars, this allows fusion to self-regulate; if fusion gets out of hand, the core will heat up, expand and cool, thereby slowing fusion, and gravity will compress it back down again, keeping fusion going at pretty much the same rate (although as mass is lost, the core gradually shrinks and so the rate of fusion slowly increases to keep it the same, so the sun will slowly heat up as fusion goes on, on the scale of billions of years). Reactors on Earth don't have this option, and so they need to use magnetic fields to keep the reaction compressed.