The great majority of springs issue cool water, because the underground is cool. We know from exploring caves that the temperature underground represents the year-round average of the region, and so it is with groundwater.
But a few springs have water of higher temperature. Geologists classify them all, from lukewarm to boiling, as thermal springs.
Warm Spring or Hot Spring?
Definitions vary, as you might expect. The big AGI Glossary uses the traditional one that a thermal spring is "appreciably" warmer than the local average temperature. It puts the boundary between a warm spring and a hot spring at body temperature (37°C). That's as good a criterion as any, and it's simple to apply.
The National Oceanic and Atmospheric Administration (NOAA), which last issued its master list of U.S. thermal springs in 1980, puts the cutoff between a thermal and nonthermal spring at 20°C, or 68°F. It doesn't draw a line between warm and hot. Often the only data it has about a spring is "warm" or "hot," though. Let's presume that those refer to body temperature too.
Western Thermal Springs: Hydrothermal Springs
The 1980 NOAA list includes 1702 thermal springs and NOAA's server at the National Geophysical Data Center lists 1661. They are overwhelmingly located in the 13 states of the West, including Alaska and Hawaii, plus nine more in West Texas and a pair in the Black Hills of South Dakota, which are geologically similar. The NGDC has a handy viewer, too.
The West is highly active in terms of both volcanoes and tectonics, one providing heat and the other one motion. Intruding magma produces the hottest springs. These go with the geysers, lava flows and other volcanic features at Yellowstone, the Cascades and many other places. Tectonic stretching of the crust and lithosphere is another way that deep superheat can approach the surface. Whether the setting is magmatic or tectonic, fractured rock is necessary too, to let water carry this heat to a spring. My article "About Geothermal Energy" gives more detail. Some researchers argue that heat caused by motion of the rocks along earthquake faults can heat springs. Drilling studies into active faults have measured heat produced that way, but I think this would be hard to prove for a particular spring.
Most Western hot springs involve hydrothermal activity, a big word that simply means water hotter than 50°C, which is chemically active enough to do interesting things (see some in the Hydrothermal Features gallery). Springs that are cool enough to bathe in are blends of hydrothermal water and ordinary groundwater.
Eastern Thermal Springs: Geothermal Springs
Outside the West there are just 34 thermal springs, of which only three qualify as hot springs (those are Hot Springs, Arkansas; Hot Springs, North Carolina; and Hot Springs, Virginia). Nearly all of these run in a string down the central Blue Ridge province of the Appalachian mountain chain. That includes the cluster in Arkansas, where the Ouachita Mountains are a continuation of the Appalachians, and likewise for the single warm springs in eastern New York and western Massachusetts in the Taconic foldbelt.
There are no active volcanoes and little tectonism east of the Rockies, so these thermal springs rely upon the ordinary heat of the Earth's crust. The crust gets about 1 degree C warmer with each 100 meters of depth, roughly speaking and as a global average; that relation is called the geotherm. The key to getting warm water to the surface is taking a shortcut across the geotherm. I count three ways to do this that favor the Blue Ridge: deep, relatively warm rock is uplifted, erosion cuts deeply into the crust allowing hot water to flow sideways to the surface, and fractures in the rock allow deep water to move quickly. Fracturing is what enables a few other warm springs to emerge outside the Blue Ridge.
The two warm springs of low-lying Florida seem to stand out as oddballs. They are Little Salt Spring and Warm Mineral Spring, near the Gulf coast in Sarasota County. U.S. Geological Survey hydrologists used chemical evidence to trace these waters to a salty aquifer (groundwater layer) about 400 meters down where the normal ground temperature, about 31°C, matches that of the surface springs. They wrote that "a preferential conduit for flow, such as a fault or fracture plane, must be present in the subsurface" to bring this water up before it can cool down. There's that geotherm shortcut again.
PS: Cold springs are appreciably cooler than normal. The way to account for them is through a natural setup that collects wintertime water and keeps it away from seasonal influences. I leave that as an exercise for the reader, mainly because I haven't seen any research into cold springs.