A broad term. Muscles can be built, enlarged, strengthened and conditioned in many different ways and although (as we have said before) any kind of weight lifting will encourage and promote strength and improvement on some level, there is usually something specific that a person has in mind.
Strength. Powerlifting, Olympic Lifts, Strongman Competitions, this is one end of the scale. Weight training here usually centres around the basic big compound lifts (Squat, Bench Press, Deadlift, Overhead Press) and uses heavyr weights with low repetitions. This trains your central nervous system (CNS) and encourages the muscles fibres themselves to grow and handle bigger weights. Expolosive lifting also plays a part, especially with the Olympic Lifts. The weight bench, barbell and power cage are your freinds here.
Size. Bodybuilding, sculpting this is what most people think of when you mention weight training. Lifting weights that allow a set of 8-12 repetitions is the range where size growth is promoted here, which usually means medium to heavy weights, less so than in strength training. This not only promotes some muscle fibre growth abut also encourages a specific fluid increase within the muscle which would carry essential proteins and energy for the muscle to use. Lifts would include the basic compound movements and also isolating smaller specific muscle groups to be worked (or sculpted) independantly. It's worth noting at this point that muscle size does not always mean muscle strength. Unless your an elite lifter who does this for a living and has all day, every day to workout, eat and sleep of course. It may look like a champion body builder has the biggest muscles in the world so therefore must be the strongest person in the world too, but not so - even at this level, a top bodybuilder may have a deadlift of around 300 to 350 kg (still very impresive) but the equivalent elite strongman or powerlifter would be lifting nearer 500kg and their body would look totally different. Weight benches, barbells, dumbbells and cable operated machines are all utilised in the pursuit of muscle size.
Endurance. Athletics, sports persons and the like will gain most benefit from this type training. If you want to be stronger than you would normally be, also fitter and with greater endurance for repetetive actions in your given sport (lets say a tennis player repeatedly swiping their racket whilst constantly shifting position on their feet) then training in the higher rep ranges will help you to succeed. Lighter weights with sets containing reps of 12 to 20+ will encourage your muscles to adapt without increasing in size too much, most athletic sports persons are not interested in carrying extra bulk or weight. Again, a basic weight set, or multi gym will be beneficial and additionally some cardio equipment such as a cross trainer or treadmill also.
So, the first thing you want to do before undertaking any weight lifting regime is decide on what you want to acheive, more commonly referred to as your "goal". Once this has been established then you will be looking to choose one of the three routes outlined above, or somewhere inbetween one and another. For example, if you are interested in both size and strength (and you don't have the time or resource to live the life of an elite bodybuilder) then a mix of both lifting styles in a rep range somewhere between the two (perhaps 6 to 8 reps per set) may be of benefit.
But, this is only one piece of the puzzle. Lifting the weights plays a big part, but equally important are nutrition and also rest. Remove just one of these other factors and the results that you are expecting will be much harder to achieve.