According to Wikipedia:
A Beveridge curve, or UV-curve, is a graphical representation of the relationship between unemployment and the job vacancy rate (the number of unfilled jobs expressed as a proportion of the labor force). It typically has vacancies on the vertical axis and unemployment on the horizontal. …
If it moves outwards over time, then a given level of vacancies would be associated with higher and higher levels of unemployment, which would imply decreasing efficiency in the labour market. Inefficient labour markets are due to mismatches between available jobs and the unemployed and an immobile labour force.
This is pretty straightforward. Companies attempt to fill vacant jobs, and as unemployment goes down (so, fewer potential candidates for those jobs), this task becomes more difficult and we move up to the left on the curve. Similarly, as there is less demand for employees (fewer vacant jobs), the unemployment rate increases (at the very least by the rate of growth of the labour force).
I just recently learned of this curve in a letter by Cumberland Advisors which has been helpfully charting the Beveridge Curve from December 2000 to February 2012, using U6 unemployment, which is a comprehensive metric that encompasses those who are working part time for economic reasons, and those who are "marginally attached" or otherwise have become discouraged and stop looking but still want work (both of these groups are ignored in the Official or U3 unemployment rate).
Here's the chart from Cumberland Advisors (source PDF): 
From the onset of the recession this chart shows the US labour market becoming less efficient. But notice that beginning last Fall the direction of the curve made a clear jump leftward after muddling through for about a year. This is encouraging. What should be clear from this is that the worst is largely behind us (look at the massive jumps during the red recessionary section) but we are moving in the right direction.