Export selected columns (to excel or dbf) in ArcGIS Desktop 10.1

You can turn Fields off by right-clicking on them in the table view and selecting "Turn Field off".

Or you can do it in the dataset properties -> "Fields" tab by unchecking the checkboxes next to the field name.

Fields that are turned off won´t be exported.

Use can use the ArcToolbox > Data Management Tools > Fields > Delete Field geoprocessing tool. Your shapefile can be specified in the Input Table parameter. Then just choose what fields will be deleted. Remember to create a copy of the shapefile first, so you have original data!

There have been updates since this question was last posted. But just in case, here is a possible way to resolve and/or split your full database table into smaller feature classes or tables.

right click on the shapefile/table/feature class that has your entire data, select the fields tab, deselect/Deselect fields for the sub class or table you need, click OK, then export your data into the format you need. It should only have the columns you have chosen

Layer > Fields > "choose which fields will be visible" > OK
then Layer>Export Data