Is it possible to increase the resolution of an image (or increase the quality of an image) using some simple software or process? If yes, how might I go about it?
Something like this: SmillaEnlarger
Will do it without just making a blurry mess too. There are other fractal enlargers out there which try to mimic missing detail - but this one is free.
As Mike says, you can only work with the information that already exists in the graphic file, but that doesn't mean you can't increase the resolution, merely that you will have to make some trade-offs to do it. sometimes you are given an image that doesn't have the resolution needed for the purpose at hand, but have no choice in the matter. In those cases you can use the software like the one linked above (which increases the pixel count and averages the original pixels on either side) to reduce the apparent amount of blurriness to some extent. Then use existing techniques that can be found elsewhere on the web to increase sharpness and make the image appear sharper than it actually is. The result will definitely be less effective than using an image of the proper resolution, but might bring the image you are stuck with at least into the realm of usability.
Mike, I was taught the same absolutes as you, but sometimes it pays to step back from the instinctive, knee-jerk reactions, and check a little deeper. This is where learning and growing as a craftsman occurs.
There's a nice online one here http://waifu2x.udp.jp/ which uses "Deep Convolutional Neural Networks". It's especially good for enlarging logos and line art.
This little free ware utility http://sourceforge.net/projects/imageenlarger/ does a good job of enlarging without the usual jaggies you get. The interface isn't the most intuitive but if you can work it out it does the job.
Sorry everyone - but you CAN NOT increase resolution on something. Why do people think you can? If the information is not there in the image - how is photoshop or something going to find it and add it in???? IT CAN'T. It's a give and take relationship that resolution and physical size have. You got something at 8.5x11 (physical) and 300dpi - you blow it up to 14x16 - guess what, it's going to be way less than 300dpi not matter what you try to increase. Vice versa it will work great. Basically You can take away but you can't add to. That's the real truth. Do your tests and prove it to yourself.
Don't answer questions you don't understand. All these "experts" out there. Just cause you own a toothbrush, doesn't make you a dentist.
Again - something that is 72dpi at 7inches will most likely be around 37dpi at 14inches. Give and take. Start your original as high as you can (be it a scan or photo) and keep it b/c that's as large and as good looking as it will ever be.