Muscle failure is the point at which your muscles are no longer effective in the movement you are trying to train them in and is a term that is used commonly in athletic pursuits and specifically bodybuilding. For example, if you are doing bicep curls with a heavy weight then you will repeat the movement for several repetitions. Once you have done this a certain number of times however, if you are using a heavy enough weight, then you won't be able to perform any more repetitions or to completely contract your muscle thereby leaving your arm unable to lift the weight. This then is called training to failure, and it is a good mark of 'intensity' in a workout. When you are training your muscles, the desire is that you go at a certain level of intensity so that you are putting in 100% effort and so that your muscles are getting an at least adequate amount of training.
So that is what muscle failure looks like, but what is actually happening that causes this? Essentially during any movement you will be recruiting muscle fibres to contract your muscles. These fibres are classified as Type I, Type IIA and Type IIB which corresponds to their 'explosive' power versus their endurance. Over time the muscle fibres will tire out and you will no longer be able to move them.
However the point of training to failure is that your muscle fibres will kick in a particular order. That is to say that your Type I muscle fibres will do all the work, and that the Type IIA and Type IIB will only kick in once those tire out. In other words, if you do not train to failure then you will likely only be training the one type of muscle fibre. This will then result in only that type of muscle fibre tearing and that will result in only those fibres growing. Training to failure = more muscle used = more muscle mass gained.
How to Train to Failure and Beyond
If you want to build muscle fast, then most bodybuilders will tell you for this reason that you need to train not only to failure, but past failure, to ensure that you utilise each and every muscle fibre. To do this you need to have a spotter, or need to devise a way to 'spot yourself'. To do this you then need to find a way to cheat and to get help from elsewhere once you reach failure. At this point you use a friend, another arm, or momentum in order to take the place of the muscle fibres that have given out. So you would train to failure and then use your other arm to help curl the weight, or you would train to failure and use a spotter to help you lift the extra weight. This way you make sure to train the muscle 100% and you will ensure the greatest growth as a result.