The general process of combining atomic nuclei is called, straightforwardly, nuclear fusion. Multiple sets of reactions can lead from hydrogen to helium, but the most common sequence, occurring in stars, is called a proton-proton chain. This sequence starts from the most common isotope of hydrogen (H with a nucleus of only a single proton) and results in stable helium nuclei (2 protons + 2 neutrons). The sequence proceeds as follows.
Step 1: Two pairs of protons combine to form two deuterium atoms (+ released energy and 2 positrons). Tracking all of the particles involved: 4H+ -> 2 deuterium + 2 positrons ( Deuterium = hydrogen with a nucleus of one proton and one neutron instead of only a single proton; positrons are essentially positive electrons. )
Step 2: Each deuterium atom combines with another proton to form a helium-3 atom; 2 deuterium + 2H+ -> 2He (+ released energy).
Note that while each produced He has 2 protons in its nucleus (as they must, since the number of protons dictates the element), each has only 1 neutron in its nucleus instead of the 2 in the most stable isotope of He. Thus, these are called He-3 atoms (3 because the nucleus has 2 protons + 1 neutron = 3 nucleons).
Step 2 alternate: The 2 deuteriums combine to form a single, stable He-4 (He with 2 protons + 2 neutrons, the most stable isotope), and released energy. This alternate step seems to be less likely than the "main" Step 2 simply because the number of free protons is much greater than the number of deuterium atoms, such that the "deuterium + proton" reaction occurs more frequently.
Step 3: The two He-3 atoms combine to form a single beryllium-6 atom; He-3 + He-3 -> Be-6 + released energy. (6 because 4 protons + 2 neutrons = 6 nucleons).
Step 4: The Be-6 atom, which is unstable, breaks apart into a stable helium atom and 2 protons; Be-6 -> He + 2H+ + released energy.
This page describes two other paths which occur after Step 2, but with less frequency.
Some other sets of reactions which produce helium from hydrogen are deuterium-deuterium, where 2 deuterium combine to form a helium-3 and a free neutron; and deuterium-tritium, where one deuterium and one tritium combine into a single helium and a free neutron (tritium = hydrogen isotope with 2 neutrons in the nucleus).
As a slight extension of the answer, a primary difficulty in using the proton-proton chain for producing power from fusion on Earth is getting the charged nuclei close enough together to fuse. Within stars, gravity greatly increases the density such that relatively low energies (often given as temperatures) are required to surpass the electrostatic repulsion between the same-charged nuclei. However, gravity on Earth is much lower, meaning much larger energies (temperatures) are required to force the nuclei close enough to combine. The temperatures required are ~6x greater than in the core of the sun, which far exceeds the capabilities of known materials.
More promising terrestrial fusion schemes are actually the deuterium-deuterium and deuterium-tritium reactions, with those reactions being 1024 times more reactive than standard hydrogen and easier to contain than the proton-proton chain. Research to enable appropriate conditions for these reactions has focused on magnetic and laser inertial confinement. And despite the theoretical benefits to nuclear fusion reactors, there are still potential drawbacks, as explored in this (lengthy) article.
[The Stellar Physics section in Unit 2 of this class website has several helpful pages. These two questions on ScienceLine are related to this one and may interest readers.]