Main difference:
Singlemode fiber has a lower power loss characteristic than multimode fiber, which means light can travel longer distances through it than it can through multimode fiber. Not surprising, the optics required to drive singlemode fiber are way more expensive, especially considering any varying circumstances.
When to use each:
Both singlemode and modern multimode fiber can handle 10G speeds. The most important thing to consider is the distance requirement. Within a data center, it's typical to use multimode which can get you 300-400 meters. If you have very long runs or are connecting over longer distance, single mode can get you 10km, 40km, 80km, and even farther - you just need to use the appropriate optic for the distance required, and again, the prices go up accordingly.
Compatibility issues:
They are not compatible. You cannot mix multimode and singlemode fiber between two endpoints. The optics are not compatible either.
There's a lot more to say about fiber, in general, but I hope this answers your immediate question.
ZR optics are spec'ed to 80km but they're not part of the 10GE standard; they're rather a "Cisco special."
In terms of what actually allows them to transmit light over this distance may be "protected" information from Cisco (both the ZR and ER operate on 1550nm SMF), but they've published some of the optical parameters for the ZR XENPAK here.
EDIT: While I thought that 10GBASE-ZR was Cisco-specific, it's not actually. The wikipedia page for the 10Gig standard states "multiple manufacturers" have introduced "ZR" optics which are based on the 80km PHY described in the OC192 SONET specs.
Best Answer
10BaseT and 10BaseFx are standards of connectivity for Ethernet.
10BaseT defines the use of twisted pair cabling (UTP / STP) for Ethernet, that includes cable features, male (in cable) and female (in wall boxes) connectors, max distances, etc.
10BaseFx defines the use of fiber optics for Ethernet, including type of fiber, max distances, connectors (in cable and in wall boxes), etc.