I'm was having the same problem, lynnie. I don't know if it's related or not, but I had installed an Add-in program called FS Pro 2013. It's an awesome program that automates the formatting of certain types of documents, but there must be something in the program macros that interferes with Word 2010's ability to tab in a table. When I disabled the Add-in, I could tab in the table as normal.
Try going to File > Options > Add-ins. Then at the bottom of the resulting screen, you're see Manage: COM Add-ins. Click on the Go button and see if you have any add-ins that you recently installed that may be interfering. Deselect everything except the add-in you suspect could be causing the problem and then click on Remove.
Hope this helps.
JP