The lower back muscles, or spinal erectors, are very important to weight train as they support or assist in almost every weight training exercise. Most upper body exercises require lower back strength to keep your body straight when performing standing or seated weight training exercises, or keep your body in a proper position while performing upper back exercises like bent over barbell rows. Lower back strength is also required to do heavy leg exercises like squats and hack squats.
The best lower back weight training exercise is the deadlift. Deadlifting is much more than just lifting a weight off the floor and getting your body into a standing position. To properly perform deadlifts in your lower back weight training routine start by using a shoulder width grip on the bar with an overhand grip with one hand and an underhand grip with the other and your legs positioned inside from your arms. Squat down and arch, not bend, your lower back and use a combination of the leg hip muscles to get the barbell off the floor. Continue upward, using the lower back to finish the deadlift, until you're in a standing position with the bar in your hands. Hold the weight for a second and lower it under control.
You also need to assist your grip in some way while deadlifting or your grip strength will give out to soon when you're deadlifting heavy. The best way to assist your grip while performing deadlifts in your lower back weight training workout is to use chalk on your hands. Using wrist straps should be avoided while deadlifting heavy because you could easily pull a forearm muscle using them. Some weight trainers, usually powerlifters, put baby powder on their thighs to help 'slide' the weight up to finish a deadlift rep. This is fine if you're training for a powerlifting competition, but it can be messy.
You could also try sumo style deadlifts during your lower back weight training workout. They are performed the same way as regular deadlifts but with your legs in a wider stance outside your arms and a closer width grip on the bar.
Another great lower back weight lifting exercise is the good morning. To perform good mornings you stand with a weight on your upper back with your feet placed at a normal standing width and your knees bent sightly. Then you bend forward, using the hips more than the lower back (keep your lower back arched) to about a 45 degree angle and go back to a standing position. Don't go to heavy or twist your body while performing good mornings. Use this as a secondary or alternative exercise to deadlifts during your lower back weight training workout.
To finish a lower back weight training routine you should do hyper-extensions. Do these on a hyper-extension machine with your upper legs on the pad and your body parellel to the floor (or at angle, depending on the apparatus), facing the floor. Bend towards the floor at the waist, stopping short at being perpendicular to the floor, and raise your body back up. You should use either no, or very light, weights while doing hyper-extensions and make sure not to twist your upper body at all. Be careful while performing hyper-extensions for your lower back to avoid injuries.
While strengthening your lower back will help you gain strength on many other lifts in your weight training routine because you'll be able to support heavier weights, it's vitally important not to overtrain it. There are many weight training exercises (squats in particular) that will suffer if your lower back is still sore from it's last workout.