Most wind instruments require the player to do something very specific with the lips and the facial muscles while blowing, in order to get a good, controlled sound. (Brass instruments will get no sound at all unless the lips are buzzing against each other and the mouthpiece.) The formal term for what a player does with the lips and face is embouchure; the informal term is chops.
Unless they are slurred, notes played on wind instruments are tongued. This means that the tongue, which has temporarily blocked or interrupted the airstream, begins each note by releasing the airstream again. Tonguing is usually done with the tip of the tongue, as if the player is saying "tah". But sometimes, when the music is very fast, some wind players will double tongue (tah-kah-tah-kah) or triple tongue (tah-kah-tah tah-kah-tah) the notes, using the back as well as the front of the tongue. Flutes can also get an effect called flutter tongue by using an articulation that resembles the rolled Spanish "rr".
In the meantime, the fingers are usually involved in making the column of air in the instrument shorter or longer, to make the pitch higher or lower. This may involve a sliding section of the instrument (as in a trombone), or fingerholes that can be covered or uncovered with the fingers (as in recorders).
In most modern instruments, however, it usually involves either keys or valves. The fingering of a note is the keys or valves that need to be held down for that note. But most instruments can get more than one note with the same fingering, by changing the embouchure to get different harmonics of the standing wave. In fact, brass winds can get so many different harmonics with one fingering that changing the embouchure is the main way to play the instrument. Brass usually use valves, and woodwinds usually use keys. Keys and valves work in fundamentally different ways.
That vibrating standing-wave column of air inside the instrument generally ends at the first place where air can escape from the instrument. So (this is simplified for explanation purposes), the more fingers a recorder player is holding down, the longer the column of air and the lower the pitch. But it can be difficult (on some large instruments, impossible) to completely cover all the holes with the fingers, so most modern woodwind instruments use keys instead. The fingers press down the keys, and the keys cover the holes as needed, usually with a pad that covers the hole more completely than a finger could, and sometimes also using a lever that lets the finger press in one easy-to-reach spot, while the lever presses the pad over a hole in a more-difficult-to-reach spot.
Valves are more commonly found on brass instruments. Pressing a valve makes the air flow through an extra section of tube, temporarily making the instrument longer in between the mouthpiece and the bell. The slightly longer instrument gets a slightly lower fundamental harmonic, and a lower harmonic series. (A few valves are ascending valves, which cut off a section of tubing and so raise the pitch.) Press the button in this animation to see how the air gets redirected through one type of (descending) valve.
The figure and the animation show one type of piston valve. Other styles of valves, including rotary valves as well as other types of piston valves, have different arrangements for the air flow inside the valve, but the purpose is always to redirect the air when the valve is pressed, opening up or cutting off a section of tubing.
Most brass instruments can play an entire chromatic scale with just a few valves. They use small changes in the embouchure to get many different notes from the harmonic series for each valve. But woodwinds have many more keys and fingerings available. Typically a woodwind can play the notes in an entire octave just by changing fingerings. Then a large change in the airstream and embouchure is needed to switch to the next harmonic, so that the next octave can be played. This big change is called overblowing.
Some brass instruments may also have a spit valve, a small hole that is normally closed but that the player can open quickly with a small key. This is not used while playing the instrument. It is used to empty the instrument of what players call "spit". Water vapor from the warm, moist breath of the player condenses in the instrument, especially when it is cold. (And, yes, there's probably a little actual spit in it, too, but not much). This can cause a bubbling sound in the tone. The spit valve is placed at a spot where the water naturally accumulates (due to gravity), giving the player a way to quickly empty the instrument during rests.