And therein lies the problem. We all had that shiny new NES that could do no wrong for a while, but then things started to go south. You'd get that telltale blinking blue screen, letting you know that something was amiss.
Somewhere along the way everyone somehow knew to blow in the cartridge, try again, and it would work. We shouldn't have done that, we were only making things worse, but the promise of playing Mario now was too tempting for our childhood minds. But even that would fail you eventually, and some of you may have an NES that just stopped working.
Here's what actually happened: The 72-pin connector in your NES that the cartridges connect into wasn't made well. Over time the contacts loosened their grip and spread out. When you blew into the cartridge, you weren't cleaning out dust-- you were depositing some moisture from your breath, which would help complete the circuit on the contacts. Unfortunately, it also corroded the contacts on the games and the connector, which only compounded the problem.
You hear that? YOU BROKE IT! That's OK, though. We can fix it. It should be obvious, but I'll just say here that you do this at your own risk, and I take no responsibility for your NES becoming irreparably damaged or setting your mother on fire.
You can try to clean and reposition the pins on the connector, but that's tedious and may not work. It's possible, I've done it, but do yourself a favor and just buy a new one: http://www.amazon.com/NES-Connector-Packaging-Nintendo-8-Bit/dp/B000A3IA0Y
Rather than typing out all the steps, this guide on iFixIt is the best I could come across. I think this is the one I followed the first time. Don't be intimidated-- this is easier than it looks. Plus, if your NES isn't working, you have nothing to lose: http://www.ifixit.com/Guide/Replacing+Nintendo+Entertainment+System+72-PIN+Connector/3822/1
Beware, after the new connector is installed, it's going to be a tight fit getting your games in for a little while. Don't force them! Just slowly apply firm and even pressure. This goes away after you use it a few times.
While you're DIYing already, you may as well clean your games of all that spit corrosion you've given them. This video describes it well, however, I would skip the part about the brass cleaner. Some people swear by it, but some others say it can cause damage. The eraser method works, just dust the eraser flakes off well before you use the game: http://bcove.me/8z24oz93
There are many other ways to do this, just search around for them. Be sure you confirm any how-to's you read. Check the comments sections and make sure the methods show up in other how-to's. There are some bad methods mentioned in some places that can do more damage to your games. You don't need to take your cartridges apart, but it does make it easier. To do so, you'll need a "gamebit" screwdriver: http://www.amazon.com/Steel-3-8mm-Screwdriver-Security-Nintendo-Games/dp/B000F8GWH2/ref=pd_bxgy_vg_text_y
Now, you should be able to play your NES like new, assuming the connection was the problem. Now stop blowing spit into your games, and start playing them!
If you want to get crazy and try some other repairs, IFixIt has a whole section on this: http://www.ifixit.com/Device/Nintendo_Entertainment_System
An easier, yet not as thorough way to clean cartridges: http://retrowaretv.com/the-dos-and-donts-of-game-cartridge-cleaning/
Replace the batteries in those games like The Legend of Zelda that save your game: http://www.thejobbitt.com/nes-guides/nintendo-nes-cart-battery-replacement-guide/
Another method to replace the battery, solderless: http://www.gamefaqs.com/nes/916386-nes/faqs/41464