Reliabilism is a form of epistemology that believes that a person has a justified belief if that belief has been attained via a reliable process. Hence the name. Seeing as this statement sounds either profoundly obvious or entirely ridiculous, let's explain a little bit more.
Epistemology (the study of how we come to know things) can essentially be divided into two camps: internalism and externalism. Internalism says that all we need to experience, perceive, and learn about the world comes from within us, while externalism holds that outside agents are required for us to perceive and gain knowledge. (I'm sure you can guess that this is a gross oversimplification of the many debates still ongoing on this subject. But that's the gist of it.)
Reliablism, by "relying" on outside processes, clearly falls under the externalism camp. It's kind of sneaky about it, though, because it shies away from out and out externalism. What it says is that if you look down the street and see a car coming towards you, that you can have a justified belief that there is a car coming towards you because you consider the neurological process of visual perception and cognitive identification to be a reliable one.
One of the nice things about this process and epistemological leaning is that its verifiable by others. For example, if a car is coming down the road, and you have a friend with you, you could ask them to write down what it is they see coming down the road, and you'll do the same. If you both wrote down "car", so much the better. In this way, reliabilism is an inductive process, and we can even assign probabilistic values to the reliabilities of various processes. Guessing the temperature? Somewhat reliable. Reading a thermometer? Much more reliable. In this way, we can also explain a reality that does not necessarily jibe with our internal perceptions of things. If you think you see Reese Witherspoon on a late-night TV show, but then the host introduces her as "Jessica Simpson", you will likely defer to the more reliable source (the host) despite your own internal process.
The theory was first coined by Frank Ramsey in 1929, ignored for 50 years, and then expanded upon greatly by Marshall Swain and Alvin Goldman. It's gained a little ground in some circles, but internalists have a lot of issues with reliabilism. The major roadblock is that there is still a gap between a belief (and even its application) and knowledge of the belief. For example, a dog can be taught to bark when it hears the word "speak", but does it really "know" that it is speaking? Along the same objection, if you ask a calculator to multiply 936 and 715 and get 669,240, you will believe this and it is true. However, what if the fact was that the calculator always responds "669,240" to every question? The knowledge you acquired was through a faulty and unreliable method, yet is still true. This seems to go against the idea of reliabilism. Reliabilists would argue that if that was the only question you had ever asked that specific calculator, then it is not a reliable method of divining math questions - yet. Soon internalists are complaining that you must break down your processes into ever more granular aspects in order to assign reliability. And so on and so on ..