Is there any Open Source GIS Software for Android in which we can perform Various Geoprocessing Operations (like Buffering,union and intersection) and that should be customizable eg: what we do in QGIS which is Desktop Based.
There was a QGIS Android project as part of this year's Google Summer of Code. Marco Bernasocchi was the author, and although it is largely a work in progress, he has managed to get the basics working on an Android tablet. Just search QGIS Android and you'll come up with several links to mailing list posts, his blog, and download links.
For a proper useable GIS with groprocessing, this is your best (rather, only) option, but it is in the very early stages so you will have to wait until Marco gets more time/funding, or someone carries on development.
I have recently been pursuing research in the same area as you, and have found that there are not currently any open source mobile GIS APIs around that would be suitable to building a client application specifically supported for Android. I did come across BeeGIS but have not had an opportunity to spend any time with it.
Additionally you can refer to this question.
You can try NASA WorldWind. They just started android support with the help of QinetiQ‘s QGlobe team.
I wanted to mention anywhere GIS by rockwell spatial for android and available at google play. While not a total gis package it does allow for some functionality in data gathering. You create data forms in google forms which the app loads and integrates position gathered by smart phone gps (in my case i used bluetooth gps unit with additional app call bluetooth gps to save on data plan). Each data set gathered is then saved in the cloud. It also provides ability to take pictures as well providing a link to photo uploaded.
As a land surveyor i built a form that i used to catalog boundary evidence found in the field noting type, condition, size and other notes. It was waiting form me on my google drive when i got back to the office. A little table message and the info came right into qgis. Great support and great tutorials as well.
I have compiled a set of opensource GIS tools (Spatialite, GDAL, OGR) for Android. This is for our own 3D map visualisation engine Nutiteq 3D SDK.
It is work in progress, but e.g. Spatialite gives you quite nice set of geoprocessing APIs, derived mostly from GEOS functions: from buffers to basic Djikstra routing. Nutiteq is a development tool, not an application like QGIS. So you would need to implement UI based on your data and requirements, and learn a bit Android Java development. But there are already some developers doing quite generic GIS apps with this, where user can define own WMS, Shapefile layers etc and do spatial queries with this.
QGIS on Android is nice proof of concept, but it is very heavy (near 100MB installer package), and UI is not tablet/phone optimised at all. Nutiteq SDK graphical visualisation core is quite lite and mobile-optimised.