Why do we care?
Because we need to know if the beam is subjected to torsion in addition to flexure.
What is it?
The point in the cross section (or outside the cross section) where we can apply load and produce beam bending without twisting. Loads applied anywhere other than the shear center will produce both bending (moment) and twisting (torsion).
Generally when we think about a beam in bending, we only think about, well...the bending. But depending on how that load is applied to the cross section (i.e. is the load through the shear center), there may also be twisting.
Here are shear center locations for some typical structural sections (from AISC Design Guide 9).
Note that for doubly symmetric sections, the shear center is coincident with the centroid. In other words, the shear center is at the intersection of the axes of symmetry. In singly symmetric sections, the shear center lies somewhere along the axis of symmetry. Lucky for us, in typical civil engineering applications we very often deal with symmetric sections. (Aero folks get saddled with the harder math.)
Intuitive Approach to Estimating Shear Center
For starters, you can probably look at the two I-beam load cases above and intuitively sense that applying a load offset from the centerline will produce beam twisting.
A more quantifiable procedure is to visualize the shear flows and estimate the point about which the net moment produced by the shear flow will be zero.
For the Mathematically Inclined
Well, I could go bonanzas with the LaTex (and I may do so a bit later), but since my lunch break is coming to an end, I'll refer you to this explanation.