I recently purchased a Zeikos flash for my Nikon D3100 DSLR?

Can anyone tell me why it fires without me prompting it to fire, for example I don't test it - it just fires off flash almost at random and very frequently. There are also times that the flash does not fire when I would like it to. I'm relatively new to photography so maybe it's something I'm doing wrong but would like to know how to fix this.

Also what does i-TTL stand for? And what is the purpose of SL mode?

I-TTL and SL should be covered in the flash units user manual

Generally buying a third-party piece of camera gear comes with its problems.

I recently paid $200 for another Nikon SB600 and was happy to pay that for it. It seems that the SB600 is keeping its value even years after its introduction.

It is very typical that people new to photography look for "deals" so they do not have to pay the price for quality. Eventually, you will find it is worth the extra bit of money just to buy the equipment made by the camera maker whose camera you are using.

I would return the flash and spend the extra pennies for a Nikon SB700

I will never understand why anyone would pay top dollar to get a good quality camera like your Nikon and then "save money" by sticking on cheapo made-in-china copies of things like memory cards, lenses and flashguns. The photograph quality "chain" is only as good as the weakest link. Fit a carp flashgun to a quality camera and you get carp photographs and risk busting your camera.

It is like buying a Ferrari and "saving money" by fitting cut-price tyres and brakes, filling the tank with low grade fuel and wondering why it doesn't perform and handle well any more.

Sorry to be harsh but yes, you are doing something wrong - you bought the wrong flashgun

To fix it get a proper Nikon one