How do astronauts communicate with NASA?

Astronauts communicate with their controllers on the earth the same way airline pilots do ... by radio.
Since astronauts may be over any part of the earth at any given time, a network of ground stations is organized around the world, to communicate with the spacecraft and relay the communications to/from the Johnson Space Center near Houston TX.

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Bi-directional communications between astronauts and ground receiving stations is accomplished using communication radios that transmit and receive high-frequency radio waves; however, real time bi-directional communications becomes increasingly impractical and eventually impossible as the distance between the astronauts and the ground receiving stations increases. Let's look at a voyage to Mars as an example. High-frequency radio waves travel at the speed of light (approx. 299,792 km/sec or 186,000 miles/sec) and as such, a message can take anywhere from 3 to 22 minutes to travel between Mars and Earth (and the same length of time for a response to get back), making normal (real time) conversations with ground crew or family members impossible. That having been said, astronauts will still be able to communicate back home and eventually receive a reply within 6 to 44 minutes depending on their distance from mother earth. It's analogous to communicating with someone via regular e-mail as opposed to a Live Messenger.