While we don't have your bike in hand, or at least a video of the issue, it sounds more like your freewheel or your freehub has failed. I suspect that new cyclists might say the gears have failed. Edit: This answer actually mis-diagnosed the problem (albeit I'd argue for good reason - it was given before videos were provided!). I'm leaving it up for general information. @MaplePanda's answer is correct.
Inside the rear hub or freewheel, there is a system of pawls and ratchets. These enable you to coast, and they produce a ticking sound as you do. When you pedal forward, the pawls stick in the ratchets. This site has a diagram of a general pawl and ratchet system, reproduced below:
What likely happened is that the pawls aren't engaging the ratchets when you pedal forward. What to do about that depends on what equipment you have. Sheldon Brown's site shows the difference between a freewheel and a cassette hub:
If you have a freewheel that's threaded onto the hub, I'd just get a new one.
If you have a cassette hub and you were determined to do this yourself, you could remove the freehub body from the hub, and see what has happened to the pawls. It's possible that cleaning the pawls and adding fresh grease would restore them to working condition. How to do this depends on the hub construction, but at minimum you have to remove any endcaps from both sides of the rear wheel. These basically thread on to the ends of the axle and prevent the hub from coming apart laterally. When re-assembling, you would have to take care not to tighten the assembly too tight (i.e. to excessively pre-load the hub, which would basically squash the bearings in too tight and cause them to wear out).
Be aware that if you are unable to DIY this, a bike store might help you, but you might find that it's just as economical to get a new rear wheel. It depends on how much labor they need to do to get at the problem. Some cities may have bicycle cooperatives that are set up to let lay cyclists work on their bicycles, and you might want to take advantage of one if you can get to one.
If your bottom bracket (BB) was the issue, you should have a hard time turning the cranks in the first place or you would feel something crunchy as you turned the cranks, but you would still be able to freewheel (i.e. coast without pedaling) fine, and the chain would still transmit power to the cogs (i.e. the geared things in back; new cyclists might call the entire assembly the gears).