It is, in fact, possible to combine hydrogen and oxygen to make water, but it's a little tricky.

Oxygen is usually present as O₂ (two oxygen atoms bonded together), so in order to make water (H₂O) the two oxygen atoms in O₂ must break apart, and that takes extra energy. This energy can come from a couple places. Adding heat (like a spark) will give the necessary energy to get the reaction started so a couple O₂ molecules to break into 2 oxygen atoms and bond with H₂ to make a few water molecules.

When these molecules form, it generates more heat (it's an "exothermic" reaction), and this heat allows a few more O₂ molecules to break into oxygen atoms and form a few more molecules of water can form, generating more heat... etc.

A way to see water forming from oxygen and hydrogen in a lab (which should only be done with a science teacher supervising) is to do a "hydrogen pop" test. You can use a small upside-down test tube to collect hydrogen gas, then hold a lit match at the mouth and you'll hear a popping sound. The heat allows the hydrogen to combine with the small amount of oxygen in the air. When you look in the test tube afterwards there will be water condensed on the glass!

Yes, one can take Hydrogen and Oxygen and react them in appropriate conditions and form water vapor. This can then be condensed (by cooling) to liquid water. This is the best way to produce the most purified water that has no other ions that are normally present in water we know.