Unit testing is often talked about in software development, and is a term that I've been familiar with during my whole time writing programs. Like most software development terminology, however, it's very ill-defined, and I see confusion can often occur when people think that it's more tightly defined than it actually is.
Although I'd done plenty of unit testing before, my definitive exposure was when I started working with Kent Beck and used the Xunit family of unit testing tools. (Indeed I sometimes think a good term for this style of testing might be "xunit testing.") Unit testing also became a signature activity of ExtremeProgramming (XP), and led quickly to TestDrivenDevelopment.
There were definitional concerns about XP's use of unit testing right from the early days. I have a distinct memory of a discussion on a usenet discussion group where us XPers were berated by a testing expert for misusing the term "unit test." We asked him for his definition and he replied with something like "in the morning of my training course I cover 24 different definitions of unit test."
Despite the variations, there are some common elements. Firstly there is a notion that unit tests are low-level, focusing on a small part of the software system. Secondly unit tests are usually written these days by the programmers themselves using their regular tools - the only difference being the use of some sort of unit testing framework . Thirdly unit tests are expected to be significantly faster than other kinds of tests.
So there's some common elements, but there are also differences. One difference is what people consider to be a unit. Object-oriented design tends to treat a class as the unit, procedural or functional approaches might consider a single function as a unit. But really it's a situational thing - the team decides what makes sense to be a unit for the purposes of their understanding of the system and its testing. Although I start with the notion of the unit being a class, I often take a bunch of closely related classes and treat them as a single unit. Rarely I might take a subset of methods in a class as a unit. However you define it doesn't really matter.
A more important distinction is whether the unit you're testing should be isolated from its collaborators. Imagine you're testing an order class's price method. The price method needs to invoke some functions on the product and customer classes. If you follow the principle of collaborator isolation you don't want to use the real product or customer classes here, because a fault in the customer class would cause the order class's tests to fail. Instead you use TestDoubles for the collaborators.
But not all unit testers use this isolation. Indeed when xunit testing began in the 90's we made no attempt to isolate unless communicating with the collaborators was awkward (such as a remote credit card verification system). We didn't find it difficult to track down the actual fault, even if it caused neighboring tests to fail. So we felt isolation wasn't an issue in practice.
Indeed this lack of isolation was one of the reasons we were criticized for our use of the term "unit testing". I think that the term "unit testing" is appropriate because these tests are tests of the behavior of a single unit. We write the tests assuming everything other than that unit is working correctly.
As xunit testing became more popular in the 2000's the notion of isolation came back, at least for some people. We saw the rise of Mock Objects and frameworks to support mocking. Two schools of xunit testing developed, which I call the classic and mockist styles. Classic xunit testers don't worry about isolation but mockists do. Today I know and respect xunit testers of both styles (personally I've stayed with classic style).
Even a classic tester like myself uses test doubles when there's an awkward collaboration. They are invaluable to remove non-determinism when talking to remote services. Indeed some classicist xunit testers also argue that any collaboration with external resources, such as a database or filesystem, should use doubles. Partly this is due to non-determinism risk, partly due to speed. While I think this is a useful guideline, I don't treat using doubles for external resources as an absolute rule. If talking to the resource is stable and fast enough for you then there's no reason not to do it in your unit tests.
Recently Jay Fields came up with some useful terminology here, he defines solitary tests as those that use collaborator isolation and sociable test for those that don't. While these are related to the classicist/mockist distinction, in that classic TDDers prefer sociable tests and mockist TDDers prefer solitary tests, there is more to the classicist/mockist difference than their attitude to collaborator isolation.
The common properties of unit tests — small scope, done by the programmer herself, and fast — mean that they can be run very frequently when programming. Indeed this is one of the key characteristics of SelfTestingCode. In this situation programmers run unit tests after any change to the code. I may run unit tests several times a minute, any time I have code that's worth compiling. I do this because should I accidentally break something, I want to know right away. If I've introduced the defect with my last change it's much easier for me to spot the bug because I don't have far to look.
When you run unit tests so frequently, you may not run all the unit tests. Usually you only need to run those tests that are operating over the part of the code you're currently working on. As usual, you trade off the depth of testing with how long it takes to run the test suite. I'll call this suite the compile suite, since it's what I run whenever I think of compiling - even in an interpreted language like Ruby.
If you are using Continuous Integration you should run a test suite as part of it. It's common for this suite, which I call the commit suite, to include all the unit tests. It may also include a few BroadStackTests. As a programmer you should run this commit suite several times a day, certainly before any shared commit to version control, but also at any other time you have the opportunity - when you take a break, or have to go to a meeting. The faster the commit suite is, the more often you can run it. 
Different people have different standards for the speed of unit tests and of their test suites. David Heinemeier Hansson is happy with a compile suite that takes a few seconds and a commit suite that takes a few minutes. Gary Bernhardt finds that unbearably slow, insisting on a compile suite of around 300ms and Dan Bodart doesn't want his commit suite to be more than ten seconds
I don't think there's an absolute answer here. Personally I don't notice a difference between a compile suite that's sub-second or a few seconds. I like Kent Beck's rule of thumb that the commit suite should run in no more than ten minutes. But the real point is that your test suites should run fast enough that you're not discouraged from running them frequently enough. And frequently enough is so that when they detect a bug there's a sufficiently small amount of work to look through that you can find it quickly.
1: I say "these days" because this is certainly something that has changed due to XP. In the turn-of-the-century debates, XPers were strongly criticized for this as the common view was that programmers should never test their own code. Some shops had specialized unit testers whose entire job would be to write unit tests for code written earlier by developers. The reasons for this included: people having a conceptual blindness to testing their own code, programmers not being good testers, and it was good to have a adversarial relationship between developers and testers. The XPer view was that programmers could learn to be effective testers, at least at the unit level, and that if you involved a separate group the feedback loop that tests gave you would be hopelessly slow. Xunit played an essential role here, it was designed specifically to minimize the friction for programmers writing tests.
2: If you have tests that are useful, but take longer than you want the commit suite to run, then you should build a DeploymentPipeline and put the slower tests in a later stage of the pipeline.
Updated on Oct 24 2014 to include the mention of Field's sociable/solitary vocabulary.