I am a graduate student and a novice in GIS field. I have few GML files that contain vector polygon data and I want to see them in a map format. The problem is that I do not have access to ArcGIS/Grass GIS software installed in my machine. I am wondering if there are online GIS website where I can upload my file and visualize it. Also is there any simple and small size free/open source software that I can install on my computer?
A great GIS that has a huge amount of functionality is QGIS. It is very easy to install especially if you use the osgeo4w installer (Windows). You can use it to view your GML file too.
OSGeo is the place you want to look for OpenSource (free) GIS tools and libraries: http://www.osgeo.org/. You will find a ton of stuff here plus links to QGIS and other desktop mapping tools (scroll down to the links on the bottom right of the page).
Look into using ogr2ogr to convert your files to another format. There's a lot more support for KML and GeoJSON in the general open-source GIS world than GML.