I am playing the guitar, and I would like to make sure the chord I am playing is correct, I downloaded the chord from the Internet, and I played it. When I play the C Chord, the tuner shows me it is a G. I check every finger and every string, and I tuned every string. All the things are correct. Why does the tuner show me G??
The tuner only works for single notes, not chords.
About the best you can do with a tuner to check your chords is to play the notes one-at-a-time. You can see if you're spelling the chord correctly.
The basic chords are all triads (three notes). The C chord consists of the notes C, E, and G.
The tuner will try to interpret a chord as a single note and it will get confused. It'll usually pick the most-reinforced overtone, or a difference tone. It'll almost never tell you the root note (unless you're just playing power chords, maybe).
Added after acceptance:
I think a much better way to check your chords is to use your ears. No, really. Really listen to it as attentively as you can. Is the bass note clear? Is the high note ringing? Are most of the middle ones there? Can you hear the character (whether it's major or minor)?
You can still go through the strings (look for exercises in Arpeggios) to make sure the left-hand fingers are positioned correctly (not choking the fret, not muting adjacent strings), but you can only check "playing the chord" by playing the chord. And only your ears can do it.
At a much more advanced level you will go back and re-check your tuning (you'll call it intonation at this point). But you'll need a strobe tuner to do it for real. You'll be focusing on very minute changes in how your fingers bend the strings (like pulling the fifth up and, if possible, slacking the third) and the simple tuners with a sharp/flat/in-tune indicator just don't provide enough information.
A strobe tuner will, in fact, allow you to tune an entire chord all at once; but you must already know what the chord is and how it's built to do it.
There are several different types of electronic tuners which determine or estimate pitch using different methods.
A plucked string produces many overtones, some with the frequency closer to different notes than that of the fundamental pitch of the note. (If you want to see this in action, try finding a spectrum or spectrogram app for you computer or mobile device and how many frequencies it displays when you play a chord on your guitar.) Sometimes the overtones are much stronger than the pitch note frequency. You can't tell by just listening because the human brain can trick you into just hearing just the base pitch.
Some electronic tuners look for a peak frequency over the entire musical spectrum. Perhaps your guitar C chord included one string producing a really a strong overtone near the frequency of a G. So the tuner says G.
Some smarter electronic tuners look for sequence of overtones, and try to guess what fundamental pitch might produce that overtone sequence. A C chord on a guitar creates a ton of overtones, including those that (among other possibilities) a low G string might produce. Maybe more or stronger in sum total than a low C.