What's the difference between alpha-glucose and beta-glucose?
$\alpha$-D-glucose and $\beta$-D-glucose are stereoisomers - they differ in the 3-dimensional configuration of atoms/groups at one or more positions. 


  $\alpha$-D-glucose 





  $\beta$-D-glucose




Note that the structures are almost identical, except that in the $\alpha$ form, the $\ce{OH}$ group on the far right is down, and, in the $\beta$ form, the $\ce{OH}$ group on the far right is up.

More specifically, they are a class of stereoisomer called an anomer. Anomers are capable of interconverting in solution. All cyclic structures of monosaccharides exhibit anomeric $\alpha$ (down) and $\beta$ (up) versions. These differences occur at the anomeric acetal carbon (the only carbon with two $\ce{C-O}$ bonds.

These two forms exist because all monosaccharides also have an open-chain form with one fewer stereocenter. When the chain closes to the cyclic structure, the aldehyde or ketone carbon becomes a stereocenter, and it can do so in either configuration. One configuration is preferred ($\beta$), but both exist. 


  Open chain form of glucose:




In the presence of acid or base (although water can fulfill this role if need be), the two anomers interconvert through the open form until dynamic equilibrium is established. The mechanism below starts with $\alpha$ in the upper left and finishes with $\beta$ in the lower right. The open-chain form is in the middle.