When I pedal they keep shifting forward randomly but not moving the bike forward When I try to pedal fast or hard or just put any sort of pressure to move the pedals foward they seem to give out, randomly jerking forward and not making the bike wheel move. 

https://drive.google.com/open?id=0B_o4pSicvYyqLWdLTFh3VHp6LTA

https://drive.google.com/open?id=0B_o4pSicvYyqdm93aGJZZW55OVU

Does anyone know what it may be?
It most likely is a freewheel failure. If your bike has 7 or less cogs in the rear it almost surely is a freewheel. If it has 8 it may be freewheel or freehub. 9 or more cogs almost always a freehub.

The failure shown in the video may be caused by 2 things: 


The pawls are unable to engage because they are seized due to
corrosion or the lube/grease has dried up.
The pawls are broken or the springs that make them engage are
broken.


Photo of a freewheel showing the ratcheting mechanism components

In the first case dropping penetrating oil into the ratcheting mechanism may free the pawls, but it is not guaranteed. The correct solution is to open the ratchet, clean it thoroughly and re build it using fresh lube.

Opening a freewheel is a delicate thing, requieres either special tools or the ability to "hack" common tools without damaging the freewheel. Normally a freewheel has a lot of loose balls and tiny parts, and reassembly requires mechanic skill.

In the second case, it may be better to swap the freewheel, because they usually are very affordable. To remove a freewheel a special tool is required, a tool that normally only bike mechanics and bike shops have. The tool has splines that fit inside the center part of the freewheel and allow to spin it backwards. The tool should not be expensive though.

Consider that to properly diagnose which is causing the failure, the freewheel must be opened anyway. In some bike shops the wage they charge for "repairing" a freewheel may be higher than the price of a new one.

The same concepts apply to a freehub, since the ratcheting mechanism is very similar. However, changing a freehub is more expensive because the whole wheel has to be disassembled and re-laced.

What is the difference between a freewheel and a freehub? The location of the ratcheting mechanism. Freewheels contain a ratcheting mechanism inside the cogs, and are just screwed into the hub. 
This link shows an image of a freewheel removed from the hub

Notice the threads in the hub and the rear of the freewheel and also the splines in the freewheel. These splines are for the removal tool.

Cassettes on the contrary are a single unit with no mechanism, the ratchet instead is placed in the hub. When the hub has a ratcheting mechanism it is called a freehub. In this case the cassette slides into splines in one side of the hub and held in place by a lockring.

Photo of a cassette removed from the hub

Notice the splines that transmit movement from the cogs to the "freebody", the part of the freehub that can spin backwards.

Following the next link for a very complete guide: Park Tools' guide to removal and installation of freewheels and cassettes