The examinations which you can appear to get jobs with government sector since I am a B.E. with ECE specialization are as mentioned below:

1- BSNL JTO exams
Eligibilty is BE/BTECH PASS In IT/Electronics stream
Maximum age to apply is 30 years
BSNL advertises the vacancy each year for this posts

2- UPSC Engineering Services Exams
BE/BTECH pass in any stream
Minimum age to apply is 21 years
Maximum age to apply is 30 years
Next exam will be held in 2014 only

3- ISRO-Scientists
BE/BTECH pass with minimum 65% marks
Maximum age to apply is 35 years

4- IT Officer Scale 1 in Banks
For this you have to appear in the IBPS Specialist Officers Exams
Eligibity is BE/BTECH pass in IR/Computer/Electronics
Maximum age to apply is 35 years

5- UPSC CDS Exams
You can apply for all the 4 wings of CDS -Airforce,Navy,IMA and OTA
Maximum age to apply for Navy is 22 years and for Airforce is 23 years
Maximum age to apply for IMA is 24 years and for OTA is 25 years

6-GATE Exams
GATE exams conducted by IIT each year
If you want to apply to PSU companies that you need to have GATE scores

After qualifying in the above examinations one has to go through the selection procedure which would vary from department to department but generally consists of group discussion and personal interviews.

After this the final decision will be made by the recruiters amongst all the applicants.