This is probably the number 1 myth which keeps women from achieving their goals. Most people associate lifting weights with turning into Arnold Schwarzenegger, but this simply is not the case and it will not happen.
To get "bulky" requires an awful lot of things - but the most important thing to remember is that you have to try and get bulky! Any time you see a rippling body-builder, remember that they only look that way as a result of very specific nutrition, training patterns, recovery, many weird and wonderful supplements (testosterone being a very important one!) and above all, it doesn't happen by accident.
Training with weights (or "resistance training" to give it it's proper name) will certainly increase your muscle, but that does not mean you will get bigger. Muscle is the part of the body which burns calories, most of the bodies calories are stored as fat. Therefore: More Muscle = Less Fat. As fat is less dense then muscle, it means your body will get smaller!
The hurdle that lots of ladies come up against is that when your muscle tone increases and fat decreases you will probably feel your skin getting tighter which many people confuse with it growing, because that's what it feels like to you, but to other people you will look slimmer, more toned, a trimmer silhouette and not to mention the other benefits such as improved posture, structural balance, reduced risk of injury, and an overall increase in your functional strength.
So remember - Resistance is your friend!