We added a sunroom back in the 2009 and the breaker panel was full so the electrician installed a tandem breaker. I am looking at the diagram for our breaker box, and wondering if he installed the breaker in the slot marked 3 in the diagram, or whether he installed it in the slot marked 5-6 in the diagram. Please see the attached pictures. The reason that I am asking is that I intend to install a couple of additional tandem breakers, but I want to understand the numbering scheme. Thanks!
To answer the question stated: slot 3 has the tandem (better called "double-stuff" to borrow a term from Harper on this site) breaker in it.
The problem you have is that whoever installed it wasn't paying attention, and installed the tandem in a slot (3) where tandems aren't allowed as per the wiring diagram, which violates the panel's listing. Swapping the regular breaker in 5/6 with the tandem in 3 will fix this rather easily, though.
(What the diagram says is that slots 1-4 cannot be used for tandem or quadruplex i.e. "double stuff" breakers, while the rest of the slots can, although to install tandems on the right side of the panel in the tandem-capable slots you need to use quadruplex breakers due to the 240V circuits there.)
The tandem breaker is indeed in slot 3. A tandem breaker simply allows you to connect two breakers to a single bus terminal. This means that both circuits are on the same leg of the service, and so can only supply 120 volt loads.
The other breaker you're talking about, is a double pole breaker. A double pole breaker takes up two bus terminals, and so is connected to both legs of the service. Connecting to both legs allows you to power 120 and/or 240 volt loads (dryers, ranges, etc.).