To tune a guitar without an automatic tuner first requires that you have an "ear for music". That means that you can hear a tone a determine the relationship between that tone and another tone. "Tone deaf" people cannot tune a guitar without an automatic tuner. The first step is to set the big string to the proper tone. Then the other strings can be adjusted using the big one as a basis. Use a pitch pipe or a tuned piano to play a low "E" note. Tighten or loosen the "big E" string until it matches the pitch.
The next smaller string is "A", a 4th higher than the E string. There are two ways to tune the A string. One way is to press the fifth fret of the big E. That is the same tone the A string should be tuned to. Play the A string and tighten or loosen it until it matches the tone of the E string played at the 5th fret. Another way is to think of the 4th interval to determine the pitch for the A string. For example, the first two notes of "Here Comes the Bride" is a 4th. Sing, "Here Comes.." starting with the big E tone and tune the A string to the pitch of the word "Comes". "Here (E) Comes (A)."
Tune the next string, D, in the same manner, playing the A string at the 5th fret or using the "Here Comes .." method. Do the same to tune the next string, the G string. The E, A, D, and G strings are each the interval of a 4th apart. The next string is the second string, but it is tuned only a 3rd higher than the G string. So you have to play the G string at the 4th fret to get the note for the B string.
The final step is to tune the small 1st string that is also an E. Again, the E is a 4th higher than the B second string, so the same technique used before can be used, pressing the 5th fret of the B string to get the tone for the small E.
Once all the strings are tuned, play a chord that uses all the strings. The G chord is good for this. If you have a good ear, you may notice that the tuning isn't quite right. Go back through the process to determine if anything is off. Even a slight deviation from the correct tuning will sound bad to a trained ear. After you master this type of tuning, you will quickly be able to get it right by hearing the 4th intervals and the one G-to-B 3rd interval.
Of course, another way to tune the guitar is to use a pitch pipe to play the string tones, E, A, D, G, B, E. But you have to be careful you are tuning to the right octave. Otherwise you might be tuning a string too high, causing it to break. The previous method described is preferred for that reason.