Some herbs are easier to grow indoors than others. Generally speaking, chives, lemongrass, mint, parsley, basil, and bay leaves are easiest to grow. Of course, what one gardener has great success with another may struggle with.
An herb such as mint may actually fare better indoors than outside. That's because mint spreads so easily and can rapidly overtake a garden. By growing it inside in a pot you are controlling its growth and not allowing it to get out of control.
Perhaps the biggest benefit of growing herbs indoors is that they are near the kitchen and easily accessed for cooking.
Just like you, indoor herbs crave natural light. That's why a windowsill is a great location for an indoor herb garden. A table placed in front of a window also works well. If space is tight, consider hanging pots either from the ceiling or on a sunny wall.
Most culinary herbs are Mediterranean in origin, which means that they like at least six hours of sun each day. If you live in an area that doesn't get much sun, especially during the winter, then you may need to supplement with a full spectrum fluorescent light or halide lights.