This error occurs when iTunes is not able to talk to the Apple signing servers.
Apple deals with this problem in detail in http://support.apple.com/kb/TS3125.
Before you apply the recommendations (correct date, time and timezone / logged on as admin / latest version of iTunes installed / latest operating system updates installed / latest router and modem firmware installed / correct proxy settings / latest security software installed / security software correctly configured / and so forth), it may save you a good deal of time and effort to quickly check your hosts file to ensure that it does not contain any entries that will point iTunes to a server other than gs.apple.com.
On Windows, the hosts file can be found here: C:\Windows\System32\ drivers\etc\hosts.
On OSX, the hosts file can be found here: /private/etc/hosts.
A standard Windows hosts file should look something like this -
# Copyright (c) 1993-1999 Microsoft Corp.
# This is a sample HOSTS file used by Microsoft TCP/IP for Windows.
# This file contains the mappings of IP addresses to host names. Each
# entry should be kept on an individual line. The IP address should
# be placed in the first column followed by the corresponding host name.
# The IP address and the host name should be separated by at least one
# Additionally, comments (such as these) may be inserted on individual
# lines or following the machine name denoted by a '#' symbol.
# For example:
# 22.214.171.124 rhino.acme.com # source server
# 126.96.36.199 x.acme.com # x client host
If you have any lines that point to gs.apple.com, backup the hosts file and delete the lines (using a standard editor such as Notepad). You may need to ensure that you have admin privileges. Reboot and try the iTunes upgrade again.
If your hosts file is clean, you will have no option but to pursue the Apple recommendations - but at least you will have eliminated one of the more common causes of the error.