Ore is a resource added in 1.0. It is obtained by mining. It is the only resource which is by default not loaded into the containers.
Planets and Moons
The amount of ore on a (non-asteroid) celestial body is infinite.
The level of Ore on a celestial body can be detected using a M700 Survey Scanner and for more detailed data in a particular biome a M4435 Narrow-Band Scanner can be used. Note that a polar orbit of 25,000 m* to 1,500,000 m* is required to use the M700 scanner, this will of course vary a bit between objects but is a good estimate. To view the ore concentration on a planet right click the M700 Survey Scanner and select Toggle Overlay and then you can use the Cutoff+ and Cutoff- to only show the highest concentrations to make it easier to decide the optimal mining location. To be able to view the ore data in the Tracking Station, the scanned data needs to be sent back using an antenna. After the data is transmitted, a new icon for Resources will appear in the right panel of your screen where the Info and Orbit icons are. Click the button and then select Resources -> Ore. A more precise determination for a particular location can be made using the Surface Scanning Module on a rover while landed.
Note that ore is a "Surface" resource. So it will only appear in the Surface tab of your Resource App, even if the ore is in the ocean.
Asteroids contain a finite amount of ore. If mining an asteroid, the mass of the asteroid will be reduced by the amount of ore extracted. There is no need to use any scanners to find out the ore concentration for asteroids, its enough to be close to it or on it.
Conversion into Other Resources
Ore can be converted into liquid fuel, oxidizer, and monopropellant using the ISRU Converter and electric charge. There is no change in total mass when converting to liquid fuel and/or oxidizer, but there is when converting to monopropellant. Conversion continues to function during time warp.
The Abundance Algorithm
The exact values of your settings affect ore concentrations strongly, and this can have big effects on an individual game. So unlike most other aspects of the game, a slightly deeper grasp of the algorithm details is important here.
The abundance of ore in a biome is determined by two "coin flips" -- ie. random number draws. The first is based on your "resource abundance" slider. Each biome on every celestial body has a "1 minus this percentage" chance of having no ore at all. If that first chance passes, then there is a second coin flip to determine the average ore concentration in the biome. This second chance is based on your game difficulty level. It is important to understand that this abundance can easily also end up "less than zero" -- so in a hard game, with a 50% resource abundance setting, you may still only end up with ore in 25% of the biomes on all the celestial bodies together.
Of course, local random variations are also applied to the ore concentrations within a biome.