For every point, distance to nearest polygon of type X?

Performing a Spatial Join will do this. Right click on the point layer and choose "Joins and Relates > "Join". In the Join Data dialog box, choose "Join data from another layer based on spatial location" in the drop down. Then choose the polygon layer you want joined. Then choose the radio button that says "is closest to it". (The selections are a little different if you're joining points to lines)

This function is also available in ArcToolbox, which is supposed to give better performance with large datasets, and provides some extra functionality.

This method is available in all versions of ArcGIS, and at all license levels.

In ArcGIS Desktop 9.3, with an ArcInfo license you can use the Near tool for this. In ArcMap make a feature selection on the polygon layer for type 'X', load them into the Near tool with the points as the input feature and the polygons as a near feature and it should work.

Likewise with ArcGIS Desktop 10 and an ArcInfo license you can do the same thing with the ArcGIS 10 Near tool.

Point distance is among the free tools in the otherwise commercial ET GeoWizards extension. It works with all license levels of ArcGIS 9 and 10.

Point distance will compute the distance from points to the boundary edge of the nearest polygon. The tool does not respect layer selections or query definitions, instead it uses the underlying feature class directly. If your wetlands are a subset of your polygons, you have to export them into a separate feature class before.

The user guide suggests to convert the polygons into polylines if you need the distance to the boundary, but that seems not to be necessary -- I've just checked with a small data set.