While having an amp to increase the volume of an electric guitar makes it easier to hear the strings and tune them, it is not an essential part of the process. Electric guitars are the easiest to tune because they can be plugged directly into a tuner that displays the note being played and whether it's too high or too low. Alternatively, though it requires listening closely, an electric guitar can still be tuned by ear even without an amp.
Plug the guitar into the tuner. While most tuning devices have microphones to pick up the notes of acoustic guitars and other instruments, the un-amplified sound of an electric guitar is not loud enough to be picked up by these devices reliably. Therefore, it's essential to connect the guitar to the tuner via a standard 1/4" guitar cable.
Select a note. Some older style tuners require the selection of a specific note against which the tuner will test the input signal. Most newer tuners, however, simply display the note they "hear", making this step unnecessary.
Play the note. Tuning usually begins at the lowest string, but doesn't necessarily have to. It is best, however, to tune the strings to open notes. Play an open note on a single string and let it ring until it registers on the tuner.
Make adjustments to the pitch. When the tuner receives the signal input from the guitar, it will indicate whether the note is "sharp", too high, or "flat", too low. Because the note can change in pitch as it rings, don't adjust as the guitar rings, but stop the note, tighten or loosen the string at the tuning peg, and then play it again. Repeat until the note hits perfectly in tune.
Tune all the strings. Use the same method to tune each of the strings one at a time to the tuner.
Get a reference point. Guitarists usually start with the lowest note on the guitar and tune it to a note from another instrument or tuner. Those with good ears can tell the true intended note just by hearing it. It's important to have the right starting point, as a reference, though, because in this process the strings are tuned to each other.
Compare notes. While any notes can be used to create alternate tunings, in standard tuning, the fifth string is tuned to the A note produced by playing the sixth string at the fifth fret. Play the reference note first and, immediately afterward, play the string that's being tuned. If they are in tune they will sound as a single note. If not, there will be perceptible vibrations that form a sonic interference pattern. Observe whether the string being tuned is flat or sharp versus the reference note.
Adjust the pitch. Strike the strings again in the same order, only this time tighten or loosen the tuning peg of the string being tuned while the two notes ring out. Rather than listening to the pitch of the notes, however, focus on the vibration of the interference pattern. The closer the strings come together in pitch, the slower the interference will be. When the strings are in unison, it stops altogether.
Tune all strings. In standard tuning, EADGBE (low to high), the fifth fret of a string provides the reference note for the next higher string, except for the second string (B), which is tuned to the fourth fret of the third string (G).
If using a tuner and tuning each string individually, even slight variations from perfect pitch can result in a guitar that doesn't quite in tune. This can happen if the strings are roughly in tune, but not tuned to each other. Since the relative pitch of the string is more important than the exact to correlation to an external reference, a lot of guitarists prefer to tune just the low string to have the correct starting point and then tune the remaining strings using the second method.