Common US thermostat manufacturers include Honeywell, Robertshaw/LUX, White-Rodgers, General Controls, Johnson Controls, and Jameson. There are several others. For most purposes they can be considered interchangeable, although various installers may have their favorites. Only the "brand names" are US. They'll most likely be made in Taiwan, Mexico, China, Indonesia etc., with instructions written by someone in India, Turkistan, or S. Africa by an ESL marketing specialist, and the "US" company probably belongs - directly or indirectly - to a British Holding Conglomerate (LLC).
Nearly all thermostats work off a transformer, often located near the device being controlled. The standard voltage in the US is 24VAC, although 6, 12, and 30 volts have been used. The most common applications assume that the transformer voltage will be switched on and off to the device the thermostat is controlling; but many heat pump systems and electric (baseboard and under-floor) require a special device to switch "line voltage" (110/220 VAC in the US). These line voltage types will sometimes use an internal battery to power the temperature sensing circuit. Battery powered thermostats can generally be used on any system, but since you have to bring the switched voltage to the thermostat there's little practical reason to do so if it's a low-voltage control line.
Many thermostats use a "mercury switch" on a bimetal coil to sense temperature, and these require - as kendall notes - that the thermostat be mounted with a specific orientation and be "perfectly level." If your new thermostat is of this kind it should have had a "permanent label" noting "CONTAINS MERCURY DISPOSE PROPERLY" (an OSHA/EPA mandated label). More "modern" digital thermostats frequently use thermister sensors that shouldn't be too sensitive to orientation, although a level mounting is probably a good idea.
Thermostat wiring - what comes out of the wall - is commonly 2, 4, or 5 wires. Many "digital" thermostats require at least the 4-wire connection, since the sensing circuit must be powered independent of the controlled circuit (2 wires each). Thermostats with a battery to power the sensing circuit may be usable on a 2-wire system, but non-battery-powered digitals likely will not work. (There are exceptions.) If you've only got 2 wires, you're probably better off with a simple (ugly) mercury switch type thermostat.
Most recently offered thermostats accommodate 4 or 5 wire connection. The 5th wire allows a single thermostat to control both heating and cooling. If you have a 4-wire installation and have inadvertently connected the "furnace hot" lead to the "A/C hot" terminal, you'll get the effect described that you can make it turn on, but it will never shut itself off and you can make it turn off but it will never turn itself on.