I am using Ubuntu 13.10 in a Intel Dual Core 3 GHz processor with 2 GB of RAM and I would like to buy a 4 GB of RAM extra and what best I can use it more efficiently and how I am supposed to achieve that ?
My Plan is to run some Database Server.
Yes. But 2 will be sufficient to run Ubuntu too. Ubuntu defaults to the PAE kernel for 4Gb+ support on 32 bits so you will be fine. PAE has a maximum of 64Gb.
A database server will benefit from more memory. But it is impossible for us to answer if it runs efficiently since this is very very dependent on the amount of users, bandwidth, and a lot more. Memory is just a small facet in this.
In order to be able to utilize all 6GB of RAM with the generic Linux kernel, you will have to install a 64-bit edition of Ubuntu. 32-bit systems are only able to exploit 4GB of RAM (at most).
Another option, if you want to keep the 32-bit OS, is to use a PAE (physical address extension) kernel that will allow the OS to "see" more than 4GB of RAM, but in order for this to work, your CPU must also support PAE.
I does not matter what RAM you will buy, as long as frequencies conform to what your processor supports.