Listen for the thrum.
In the history of the guitar the automatic or digital tuner is a very recent thing.
Tuning a guitar without a tuner requires that you have an ear for music, which to my mind is nothing more than the ability to listen properly. If you do not have this ability you are probably not an ideal candidate to learn an instrument anyway.
There are several ways to tune a guitar, with experience and practice it will become second nature although the skill of having 'perfect pitch' is something that not everyone can achieve.
What do I mean by perfect pitch? It is the ability to name a note as it is played. With practice I have found that I know what a guitar's top E sounds like, I've been playing guitars for over thirty five years and have only recently found that I can do this. I recently re-strung one of my guitars and tuned it roughly to set the strings in place before reaching for the digital tuner to perfect it.
I had been playing most of the afternoon and must have had the note in my head as, when I checked the top string against the tuner it was only out by a tiny amount.
So how do you go about tuning the guitar?
Firstly you start with the bottom E string. This is the string that puts most tension on the guitar's neck. If you tune it last you run the risk of increasing the tension on the neck and putting all the other strings flat as you do so.
Secondly you always tune up to a note, never down. If the string is sharp loosen it and tune up again.
Find some way of getting the first note, either from a pitch pipe or by pressing the low E on a piano or keyboard. When I played with a band we got the keyboard player to give an E so that we could all tune up to him.
When the bottom E is in tune hold it down at the fifth fret and compare it to the open fifth string. Here's the secret: when the two notes are nearly the same you will hear a rhythmic pulsing known as beats. You can hear the same effect from multi-engined aircraft or boats if their engines aren't in perfect synch. Play the two strings together; you will hear the beats as a kind of wuh-wuh-wuh sound. As you get nearer to accurate tuning the pulsing will become faster until, when in exact tune, the sounds merge together and complement each other.
When you are happy with the fifth string fret it at the fifth fret and compare it to the fourth and so on until you get to the third string. When this is in tune fret it at the fourth fret to compare it with the open second string, then fret the second string at the fifth fret again to compare it with the open first string.
Play a chord. If you have got it right it will sound right. If it doesn't sound right do it again. It will come with practice.
The beauty of this method is that, by using the beats you only have to tune the bottom string and you can rely on the mis-synchronisation of the beats to ensure that the others are right. I find that this method is accurate enough for me and checking with a digital tuner only confirms what my ear has told me.