I try to turn on my laptop, and it won't turn on. When it originally happened I was on it, without the charger plugged in, online. It was working fine, and then all of a sudden it shut off, and will no longer turn on. The power light turns on, but the screen does not light up, nor do the fans start to run like they normally do. What is my problem?

Your battery is probably completely discharged. This can prevent the laptop from starting even WITH the power plugged in. You might try removing the battery, then plug in the AC power adapter. Push the power button and HOLD IT down for 30 seconds (your laptop may or may not start up. If it starts, then shut it down normally.)
Replace the battery. With the AC still plugged in, repeat the start proceedure above. Leave the laptop on until the battery is fully charged. If this doesn't work, then your battery may be bad (this can happen all at once) or worse, your charging system isn't working (less likely.) Your power adapter could also be bad - is the green light on when it's plugged in?

You may have to repeat the proceedure a few times before it works as the battery needs to show a little charge before it will start.

Once started, press F2 while on the dell logo to go to the setup screen. Select the power options. and you should see if the battery is detected. If battery is detected, it should say "charging".

If none of this works, call Dell technical support 800 822-8965 for further assistance

Did you try to power it up without the battery?