We hear this question all the time and without fail, the answer is the same, it depends on what type of bed you have. Elaborating on the topic, beds typically come in three styles: Headboard only, headboards with rails, and platform beds.
Headboard only beds are exactly that, Headboard only. With headboards you will need to use a standard metal frame, typically purchased through your local mattress dealer, and attach the metal frame to your headboard with screws. In this type of configuration you will need to use a boxspring. Having a headboard only will also require you to use a bed skirt.
Headboards with rails are also a style that requires you to use a box spring. The bed rails will typically have a metal support system built in (sort of like the metal frame you would use for your headboard) for the box spring but this support system is not designed to hold the mattress. No bed skirt is needed with this style bed but you will want to buy a decorative box spring cover in case a portion of the box spring peaks above the side rails and footboard.
Platform beds are the most popular style bed and tend to have the most contemporary and streamlined look. Platform beds are unlike any of the above mentioned configurations since most platform beds have a box springs built into the platform itself. Depending on the style of bed you purchase, it may still be considered a platform bed but only have slats rather then a built in platform. With these styles of platforms, your use of boxspring depends on the type of mattress that you own. Most mattresses will not require a boxspring in either of these types of platform beds.