Is it possible to install Sharepoint server 2010 on a Windows server 2008 without using SQL Server? I've installed all the other prerequisites and the installation is complete. Problem arises when I try to configure the server and create a new farm. It asks for an S

SharePoint stores its data within SQL Server, so you need a SQL Server. It doesn't have to be running on the same server as SharePoint, though.

Ok, Thank you Trevor. I would like to know one more thing, as you said I am trying to install the SQL server 2008 R2 on my system which is a domain controller, i.e Windows Server 2008. I am planning to install the Sharepoint server 2010 on the same machine. Is this  appropriate?

It is supported for 2010 (it is not in 2013 for production environments).

You will need to make sure you've got enough CPU and RAM for both and be careful that SQL doesn't steal all the RAM for the server.

Yes. You can install the SharePoint Server 2010 in the same machine.

http://www.microsoft.com/en-us/download/details.aspx?id=29073
http://technet.microsoft.com/en-us/library/cc287960%28v=office.14%29.aspx

http://technet.microsoft.com/en-us/library/cc262485%28office.14%29.aspx

http://technet.microsoft.com/en-us/library/cc990273%28v=office.14%29.aspx

While installing the SQL Server, in the Reporting Services Configuration section, I had selected native mode instead of Sharepoint integrated mode. Should I change it?

For SQL 2008 R2, you want the option to leave it unconfigured. You would then integrate it post-installation with the Reporting Services Configuration Manager, then Central Administration. But this requires you to run the SSRS instance on the SharePoint server. So if this is just your SQL Database Engine server you do not plan on installing SharePoint on, you don't need to install SSRS.