I would like to know, why it is required to reboot, to apply the changes, which I made.
I refer to the use of
adduser username sudo
I use Ubuntu 12.04.1 LTS "Server".
You are not required to reboot the system but you have to log out and log in again. Group membership is evaluated at login time only.
If you are in a situation in which logging out and back in is inconvenient (for example, an SSH session) then you can just spawn a login subshell with
to test, from your terminal call
which will show you your username and group, for your current login session. it's the only way to know for sure whether your change was correct. The output will be something like
uid=102(andrew) gid=101(users) groups=101(users),14(sysadmin)