Your name is different in your SSC or 10th certificate.This needs to be rectified so that you do not have future issues with jobs etc.
Please use this opporutinity to check for other errors in your certifcates and ID dcouments.
If the name mistake is done by the school authorities ( and you have filled in your forms correctly), then the school has to verify and get this corrected. It would be easier for you to initiate the name change
You need to have at least two documents that state your correct name ( birth certificate, 12th marks card, PAN or Driving license).
You can use attested copies of these are proof of your correct name. If you do not have this or your 10th is from a CBSE school, then you need a name correction or change affidavit made, this will be done by the Notary in your region/ civil courts. Once you do this, you would need to publish the name change in two newspapers, one English, one vernacular language ( this is not difficult, just approach any newspaper office nearby they will do it for a small charge).
You need to meet the current principal of your old school/administrative office, draft a letter request the name the service you need and then forward it to the state board/10th board via the principal or current head of the institution.
Also, enclose the old certificate copy, ID proof or name affidavit with the prescribed fees.
You should get the new corrected certificate in around 4 weeks time.
If it is the CBSE for your 10th, then the process is the same but you need a formal form and the name change affidavit, this link gives your the guidelines and the downloadable forms for the same (http://cbse.nic.in/newsite/forms.html)