The short answer is approx. 1 to 2 lbs of muscle per month if you are weightlifting daily, but there are a number of factors that affect how fast you build muscle, listed below, which will make you gain muscle faster or slower.
Training Age
Don't confuse this with your actual age. That is a different factor.
Your training age is how long you’ve been lifting weights and engaging in weight training, either as the result of working out or as part of your daily routine (eg. If you're a construction worker you are probably more used to it). If you’re new to weightlifting you will be able to gain more muscle faster than if you’ve been lifting for many years. I know that seems backwards, but that is the way it works.
Hormone Levels
Depending on the person the amount of testosterone and other muscle building hormones in your body can vary wildly because your lifting regimen does not help elicit a testosterone response (you might not be lifting with your legs, or not lifting heavy enough), or your body simply doesn’t produce as much testosterone (due to genetics / personality). Alpha male personalities tend to make more testosterone (in theory you can even become an alpha male with time if you make active efforts to become more confident and your brain will increase your testosterone levels as the result of rewiring of your system - this is harder to do as you get older as such things become hardwired in your brain while you are a teenager).
For women muscle is still governed by testosterone levels, but since women produce less testosterone other muscle building hormones play a greater role in the female body. Cortisol thus plays a bigger role for women, but too high levels of cortisol hurts bone density. Thus for women its better to use short but intense bursts of weightlifting to get the benefits.
Genetics
Everyone has basically the same genetic coding, but there are minor differences in the coding. Don't read too much into genetics because sometimes people can defy the odds through conditioning. Most people (roughly 68%) by definition are genetically average so the chances are likely you're pretty average.
There is a concept of a genetic bell curve. To summarize what this means, some people are naturally inclined to build a lot of muscle because of genetic factors which effect hormonal balance, or the thickness of their frames (big boned) allows them to add muscle easier. A person with a small frame will have trouble building muscle.
Muscle Memory
If you used to have lots of muscle when you were younger or before an injury, then you can regain that muscle faster when you start training again. So if you are an athlete and get injured and lose 30 lbs of muscle you can regain that muscle at a much faster rate than a normal person because your body remembers the muscle being there and will build it at an accelerated rate. Its possible to gain 20 lbs of muscle back in only a single month if you have the muscle memory, because your body has a mechanism for restoring the previous homeostasis.
Protein Supplements / Steroids
Consuming protein supplements and Creatine (a chemical found in red meat) will help you build muscle faster.
Heavy duty supplements like steroids are harmful and will damage your internal organs. So called performance enhancing substances can help you build muscle much faster, but will cause your internal organs like liver, kidneys, etc to shrink and become more prone to diseases/cancer. It also shrinks your testicles and damages brain matter.
The safest route is to stick to natural supplements like whey protein and Creatine.
