This answer is a complement to Meade Rhubenstein's which covered the dumbbell very well. Your question also included the barbell. There are two main types of barbells:
- "Standard" barbells, the diameter is about 1" on average, but there is no guarantee that the weights from one manufacturer will fit the bar made by another. In short, they are not really standard.
- Olympic barbells, the bar itself is about 28-29mm diameter and the sleeves for the weights are roughly 2" (the actual measurement is metric). These are designed to true specifications, and you stand a better chance of being able to mix and match bars and weights from different manufacturers.
The average weight for a standard barbell is about 10-15lb, and won't be able to hold a lot of weight. This is both due to the fact that the sleeve for the weights is too narrow, and the fact that the bar will bend if it is overloaded. They are cheaper, and the sleeves (area for the weights) do not rotate at all. They are acceptable as a starter set.
Olympic bars come in three standard sizes depending on the category they are designed for:
- Juniors: 10kg (~25lb) designed for youth weightlifting
- Womons: 15kg (~35lb) designed for women's weightlifting competitions
- Mens: 20kg (~45lb) designed for adult men's competitions
The grips range in size, but men's bars tend to be around 28mm-29.5mm depending on if the bar is designed for Olympic lifts (more spring) or power lifts (more rigid).
The more inexpensive bars do not have rotating sleeves, and really can only be used for slow lifts. Rotating sleeves helps prevent lateral torque while you are doing squats or any more dynamic lift. The inexpensive bars have all the dimensions of a competition bar, but can't bear as much weight.
Within the bars that have rotating sleeves, there are two main types:
- Bushing bars. The sleeves rotate on bushings which is more than enough for slower lifts (power lifts and most body building lifts). The bushings don't wear out and they don't require much maintenance.
- Bearing bars. The sleeves rotate on needle bearings which provide quick, free flowing rotation necessary for the more dynamic Olympic style lifts. They require more care and maintenance, but reward you with better performance.
With either type of bar, once you get into the training and competition bars, the bar strength is measured in PSI. The higher the PSI rating, the more weight the bar can handle as it moves through the air. A minimum rating for a quality barbell will be about 155K. Best for Olympic lifts would be 195K or higher. These bars will be able to hold more than 1000lbs while being used. The bars that don't publish PSI ratings and claim over 1000lbs load only measure that load with the bar in the rack.
Bottom line, if you are just starting out, you can get a decent starter kit of an Olympic style bar with 300lb of weights for roughly $300 USD (check Amazon). If you scour Craig's List, you might be able to find a really good deal. If you really get into barbell weight lifting, you can invest in a good bar when you are ready for it. Until then, you just need to by a pair of weight plates if you max out your 300lb starter kit.