Depends on just what bicycle it is, but its probably possible to add gears in some way. But it may be expensive or complicated.
First thing to check is the frame spacing, ie the width between the rear dropouts. Many singlespeeds use 120mm wide hubs, whereas modern geared hubs are 130mm (road) or 135mm (mountain bike). If it is a steel frame, then it is usually possible to spring it out a bit, up to about 5mm or so, to fit a wider hub.
If the frame is only 120mm, then your options are limited. There are some internal hub gears for 120mm frames, eg from Sram or Sturmey-Archer. But they are fairly expensive, or have a limited range of gears.
If your frame is wide enough for a 130mm or 135mm hub, then you can use a standard road or mountain bike hub with a cassette.
Then you will need to attach the derailleur. If the frame is designed for singlespeed, it won't have a derailleur hanger. But you can bolt one on, depending on what type of droputs you have. eg if you have horizontal dropouts, you could use this DMR Chain Tugs & Mech Hanger.
Also the frame probably won't have guides for the gear cable. You can buy clamp on cable guides, or you could just use zip-ties to hold the cable in place.
Whether all this is worth it is another question. As you will have to buy quite a few parts (wheel, cassette, derailleur, gear levers etc), which could get expensive. Plus quite a bit of work fitting it all. It may be easier just to buy a bike with gears instead.