The Tier system is a structure used by technical support organizations, particularly large and outsourced ones, to (ideally) provide the best type of technical service for each caller. My own experience with technical support has been in the Internet Service Provider field, although all types of tech support use this model to one degree or another.
Most Tier systems are broken down into 2 to 5 tiers, such as Tier-1, Tier-2, Tier-3, etc. Sometimes inside of an organization there will be an informal "Tier one and a half", etc.
A caller, when they ring through to a technician, will usually be talking to a Tier-1 tech. Chances are, this technician is a representative of, not an employee of the corporation the customer is trying to contact. In most organizations, the Tier-1 technician is (ideally) trained to deal with any issues that originate on the customers end. However, in some organizations, Tier 1 only provides billing information, tracking for shipped parts or the like, and Tier-2 refers to the technical support. Often, in these situations, Tier-1 and Tier-2 are working under the same roof, often inside of the same cubicle. However, Tier-1 is usually the technicians who are responsible for all normal problems originating on a customers computer.
Often Tier-1 has an inside line to a mentor line, where they can ask senior technicians for advice on particularly odd problems, such as "Whenever I open up Minesweeper, my cable modem starts beeping out the 9th Symphony". However, the Mentor Line (as it was called in my organization) is not the same as Tier-2.
The major differences between Tier-1 and Tier-2 are that Tier-2 is usually staffed with actual employees of the corporation, and that Tier-2 is meant to deal with issues that are either unfixable on the customer's computer or that can not be resolved by standard operation procedure.
For example, if the customer calls up with an Error 691, the Windows 98 password error, there is a 99% chance that this error is caused on the customer's end, either because they mistyped their passoword, or because Dial Up Networking is somehow corrupt. However, if the problem is caused by the 1% chance that the server should be authenticating them, but is not, that is a Tier-2 issue. The basic job of Tier-2 technicians is to fix any problems that can only be fixed with access to the ISPs servers. In addition, they sometimes fix issues that are murky for a Tier-1 technician, such as a computer that can make an internet connection to one ISP, but not to another. They also often solve simple problems that Tier-1 technicians could have fixed, but through some oversight did not.
Basically, then, there are two tiers, separated by the fact that one is outsourced and another is in-house. There can often be more tiers then that however, since the in-house technicians will have additional tiers to deal with really weird problems, and problems that need such fixes as line repair, dispatch of a team to a customer's house, or someone to unplug the server and clean out the dead mouse that crawled in there and died and is stinking up the works.
The thing to remember when calling tech support is that the technicians (usually) know what they are doing, and that they are designed to give you the support needed. The Tier-2 technicians usually do not have any knowledge that the Tier-1 technicians do not, they just have additional powers. However, if the problem truly is on your end (and it usually is), the Tier-1 technician, in a well-trained organization, will know how to fix the problem. For that reason, Do not ask to talk to a Tier-2 technician, unless you are absolutly certain that the problem is not on your computer. Doing so will not only waste everyone involved's time, but is somewhat rude.