Eydis,
In general terms, to get the design of the map you refer to and presuming the GPS point data contains the breeding attributes of one species, and the grid is a polygonized fabric with one point max per cell:
1. do a spatial join with the polygons as target and the points as the join table in ArcMAP (points to Polygons and select option to add all attributes)
2. The resulting output polygons shapefile with have an automatic field call "distance": Select from the new polygon shapefile those polygons with "0" distance (means they fell within) and export this selection to a new shapefile (this step is just to eliminate the grid cells with no points)
3. add 2 fields to the new join polygon shapefile and use Field Calculator to create the X and Y coordinates of the Centroid of the polygon.
4. export this shapefile to a table (dbf etc.)
5. Using the Add XY Data (or the Arc 10 equivalent Display XY ..) create a point file using the new XY coordinates of the centroids
6. symbolize the points using the breeding characteristics as Unique Categories (or create buffers) with nothing appearing for the
7. overlay this on a map of the area with or without the grid polygons
Each of those steps assumes a bit of how - to familiarity with Attribute Table operations (right click menus and table Options), but all basic and easily found in the help files as you proceed.
No doubt there may be more elegant ways of doing this in Arc10, but I think this will work for what you want. Post again if you have questions specific to any step or find something better.
Hardolph
