Angular (or Spatial) resolution is a measurement of how close two objects can be and still be seen as two separate objects. Now this is rather hard to picture in your head so lets think of a simple example. If you hold a book at a comfortable reading distance you can see all of the letters on the page as distinct objects without them blurring together into an unreadable mass; However if you move that book back farther and farther eventually your eyes can't make out the individual letters and all you see is a black blur on a page. At that distance when you can no longer see the individual letters the Angular or Spatial resolution of your eyes is no longer high enough to read the book.
So what then is Angular resolution measured in? Well, it's not as simple as pixels. From the name we can tell that it has something to do with angles and this intuition is correct. Angular resolution is measured in degrees of arc, arc minutes, and arc seconds; and as in geometry class a degree is 1/360th of a circle placed around the viewer, a minute of arc is 1/60th of a degree and an arc second is 1/60th of a minute of arc and so on. Most people need only to concern themselves with degrees and arcminutes because that's what the human eye deals with in fact the human eye has an Angular Resolution of 1 arc minute (for someone with 20/20 vision). A quick rule of thumb is that your hand held at arms length with all five fingers splayed out covers about 20 degrees of arc; likewise holding your thumb straight up and down will cover about 1 degree of arc in your field of vision.
So by now you must be wondering where these measurements are used. The most interesting use is astronomy but the measurements show up in many other things. In astronomy they build very large telescopes to increase their angular resolution so that they can resolve objects further away. When we talk about stars if our telescopes had the same resolution as the human eye the closest binary stars and the closest galaxies would all look like the same type of object! Indeed a telescope operating in optical wavelengths (the light the human eyes can perceive) need angular resolutions of less than an arc second going down to .001 arc seconds in an array telescope to see galaxies and binary stars. In radio light telescope arrays can achieve much higher resolutions because for an array of telescopes the angular resolution is calculated by the distance between the furthest antennas and it is cheaper to construct radio telescopes because they don't rely on expensive perfectly formed mirrors so larger arrays can be built.
Which brings use to our last section which is how to calculate the angular resolution of something. The formula is quite simple and it looks like R=λ/D. R is the angular resolution λ is the wavelength of the light your looking at D is either the diameter of the telescope or the maximum physical separation of any two telescopes in the array.
With the new information instilled in you by this article go forth and do science!