Is sound enabled in the operating system? In Windows, the basic "Sounds" menu is found in Control Panel. Sound events that have little speakers next to them are enabled. Enable sounds for some actions that you recognize (like "exit program" or "minimize") and see if your speakers work now. Note that operating system sounds don't need to be enabled for music CDs or games to work, but if you're here, we're starting off with the assumption that you aren't getting any sound out of the speakers at all.
Are the speakers plugged in to a power source? The volume control on speakers usually serves as an on/off as well. There are some super cheap speakers that don't offer amplification, in which case there won't be any power cable, but don't expect much in terms of volume or quality. Really ancient speakers may be powered by batteries rather than a transformer, so if your speakers don't have a power cable, double check that they don't have a hatch for batteries.
Many older sound adapters and notebook computers have a manual volume dial, as do most external speakers. Manual volume dials should be adjusted to somewhere in the middle of the range, not all the way to one extreme or the other. Did you check? Check again. You don't want to invite a friend over to help with your sound problem, only to find the volume is manually turned down.
Have you loaded the best driver for the sound card? The best driver means the most recent driver, so check the manufacturer's website for an update, because your sound card may have been sitting on a shelf for a year before you bought it. If the sound is integrated on the motherboard, check for an update at your motherboard manufacturer's web site.
Does Device Manager register the sound card or integrated motherboard audio and report that the device is working properly? Even if you don't have the proper driver, Device Manager will probably identify it as a sound controller. Before you start stripping down the system or chase off to conflict resolution, make sure the adapter is seated in the motherboard slot.
If the Device Manager reports a resource conflict between any of the audio devices and another device, look through all of your device reports and figure out where the conflict is. It may be resolvable by changing the settings in Device Manager, or it may take aggressive reshuffling of adapters. If you get a "!," "?" or "i" on the sound controller in Device Manager, proceed to Conflict Resolution.