Decided I'd go ahead and throw this one out there for anyone who is just starting with weights. This is not meant to be an exhaustive, all-inclusive undertaking on my part, but merely an attempt to explain the basics of strength training in general.
You can strength train using your bodyweight, machines, and free weights. They each have their place in training and can be used to accomplish different things. Bodyweight exercises allow for greater body control and strength within a lot of different movements, both controlled and explosive. Machines allow for specific ranges of motions and particular movements to be trained that may be hard with free weights or bodyweight exercises. Free weights allow, in general, for the greatest development of "maximal strength;" that is to say, the most amount of weight that you can possibly lift or the greatest amount of force that you can generate.
The point of strength training, as is in most types of training, is to progress. You don't want to lift the same weight for the same number of reps every single week, because you won't get any stronger. We're gonna talk about this in very simple terms. The simplest way to increase your strength is to lift or exert near maximal force during an exercise. The further away you get from that maximal force, the more you're entering into the realm of muscular conditioning and endurance work (which are both important, to be sure, but not the subject of our current discussion).
For example, say that your maximal lift is 100lbs, meaning that you can only lift it once, and you wanted to increase your strength. The number of reps you can perform with lower weights will correspond to certain percentages of this maximum lift. A good range to lift in for muscular strength is 4-8 reps, which corresponds to about 80-90% of your maximal strength. That means you'd be lifting 90lbs for 5 reps or 80lbs for 8 reps. You can do this for about 3-6 reps depending on your level of fitness. Of course there are other methods of strength training, but this is just an example.
A good progression when you can lift more according to the above percentages, is to increase your working sets by 5%. For instance, when you are performing 8 reps with the weight with which you were previously performing 4 reps, you can increase the weight for that by 5%. So, instead of doing 90lbs, you would do 94.5lbs (rounding up or down if necessary) and use that weight until you can do it for 8 reps. After several weeks of this progression, it will be time to test your strength. I'd say a good benchmark would be about 12 weeks. This is for two reasons: 1) It gives you sufficient time to progress using the training program and 2) You'll surprise yourself when you find out just how much you're able to lift.
When you're ready to test your strength, you'll want to take the the last working set that you did with 4 reps and add about 10%. If you previously lifted 105lbs for 4 reps, you'll want to use a load of about 115lbs for your one rep max. Here's an example test based on the above numbers:
Warm-up Set: Easy 10 Reps with 80lbs
Set 1: 1 Rep Max with 115lbs
Set 2: 1 Rep Max with 120lbs
Set 3: 1 Rep Max with 125lbs
The reason I have the additional sets is that if you are able to complete the maximal lift, you should attempt the a higher weight. If you fail on your first attempt, be sure to give it another go, but perhaps drop the weight slightly and work it back up.
If you're lifting weights, is always good to have a spotter, so please do so.
Hopefully, this shed a little light on maximal lifts for those of you not familiar with weight lifting. If you have any questions, leave them in the comments.
Until next time, good luck and train hard!