There are lots of Arc toolboxes for ArcGIS 10 to calculate the landscape metrics.

Such as Vector-based Landscape Analysis Tools, Basic Raster Landscape Metrics Tools, etc.

look at the links below:

http://www.arcgis.com/home/item.html?id=36f9728a895e4f5386bdec68be6d08ac

http://www.umesc.usgs.gov/dss.html

First, you should generate a thematic map of the satelite image, if you work with that type of information. In FRAGSTATS, the input must be in raster format, and you can calculate a variety of metrics, from metrics of the class of interest, to the metrics of the landscape, including other types of classes in it. You can find documentation in the web of the author, Kevin MacGarigal, http://www.umass.edu/landeco/research/fragstats/fragstats.html.

You can also estimate landscape metrics in ArcGIS, but I'm still learning. 

From the FRAGSTATS webpage that was provided in the answer above, you will want to go to the "documentation" page, and then click the link in the text which will take you to a .pdf that has the formulas for all the metrics. Here is a direct link: http://www.umass.edu/landeco/research/fragstats/documents/fragstats.help.4.2.pdf

It is not clear from your question if you simply want to run the tools to calculate landscape pattern metrics, or if you are interested in the formulas and meaning of the metrics. If you are interested in the latter, you may also want to take a look at:

Turner MG, Gardner RH, O’Neill R V. 2001. Chapter 5: Quantifying Landscape Pattern. Landscape Ecology in Theory and Practice: Pattern and Process. New York, NY, USA: Springer 

and 

Riitters KH, O’Neill RV, Hunsaker CT, Wickham JD, Yankee DH, Timmins SP, Jones KB, Jackson BL. 1995. A factor analysis of landscape pattern and structure metrics. Landsc Ecol 10:23–39.

It is important to keep in mind that the extent of your landscape and grain of your data can have a strong impact on the calculation of spatial pattern metrics, and that many metricts are highly correlated with each other.

Lastly, if you program in R there are now some packages to calculate landscape pattern metrics in that program. Google "r fragstats" for a few options.