When I used Inkscape before, I was able to snap an object so it would align with the center or edge of another object nearby without going into the Align and Distribute window. Now I can't do it anymore. Did I accidentally turn this off without knowing about it? All I've been able to find is how to snap to corners and use the align buttons.
You probably switched off one or more of the snapping options in the Snap Controls Bar.
A guess would be the "Snap bounding boxes", "Snap midpoints of bounding box edges", and "Snap to page border", or perhaps you switched off "Enable snapping" altogether.
Try setting the snapping controls as shown below (blue highlight is switched on). If you mouse over the buttons, a little tooltip appears to tell you what each snapping option does.
Here's an example of what these options do.