A probe is used to define the connection between the oscilloscope and the signal to be measured. It is also possible to use a simple cable, but then the resistance and capacity of the measurement is undefined. At low frequencies in the kilohertz range, this does not play a role. At high frequencies in the megahertz range, such a cable connection falsifies the measurement signal and leads to incorrect conclusions or unexpected effects. For this reason, oscilloscope probes were invented to measure a voltage under known conditions. On the side of the probe there is a small crocodile clip. This is the ground connection of the probe. On the top of the tip, the probe head has a metal contact. With this contact the voltage signal to be investigated is touched. The metal contact is connected to the inner BNC contact of the oscilloscope. Sometimes a cable clamp is integrated into the tip. By pushing down the tip, the cable clamp can be pulled out and a permanent connection can be made.
An oscilloscope generally has an input impedance of one Megaohm. A common passive probe head has an input resistance of nine Megohms. With such a configuration the probe together with the oscilloscope realizes a voltage divider with a total resistance of 10 Megaohm and a divider ratio of (9+1) Megaohm to 1 Megaohm = 10:1. This means that a voltage measured by a probe is attenuated by a factor 10 at oscilloscope input. The voltage value on the oscilloscope must be scaled by a factor 10 to get the actual voltage at the tip of the probe. Depending on resistance or impedance (wave impedance) of the probe different divider ratios are used. Typical ratios are 10:1, 20:1 or 100:1. High-frequency oscilloscopes have an input impedance of 50 Ohm, a 10:1 Probe then has an integrated 450 Ohm resistor because (450+50) Ohm :50 Ohm results in a divider ratio of 10:1. The defined characteristic impedance of the probe also means a defined capacity. Usually these are around some picofarad. The inductance of the probe also plays a role, however for low and medium frequencies it is not so important.
A passive probe is the simplest variant. Passive probes are inexpensive, do not use active components and require no power supply. Passive probes consist of a resistor and a capacitor connected in parallel to it and look similar to the pictures shown above. Despite the simple construction passive probes improve the detectable frequency bandwidth compared to simple measurements with BNC cables.
Passive probes are not suitable for very high frequencies. The capacity of a probe is between 10 and 30pF, which results in limited measurement bandwidths. Frequencies in the high MHz or GHz range are attenuated depending on the frequency. Since the attenuation is not known you cannot distinguish between measurement artifact and measurement signal.
In the EEVBlog you can watch an interesting video about the impedance of probes and the relationship with the bandwidth, especially with switchable probes, which can be switched from 10:1 to 1:1.
An active probe has electronics that amplifies signals with high bandwidth. An active probe therefore requires a power supply. Operational amplifiers are used in the probe, which allows a high input resistance at low capacitance. This allows them to work in higher frequency ranges. The operational amplifier in the probe limits the voltage range that can be detected. The maximum permissible voltage swing of an active probe is smaller than the voltage swing for passive probes. Active probes have additional connections next to the BNC connector. There is no standard for this, i.e. every manufacturer has its own principle for powering an active probe. This means you have to select an approved probe for your oscilloscope. More info and simulations of the active probe can be found here.
A special form of active probes are differential probes. They do not measure against the mass of the oscilloscope and prevent in many cases a short-circuit of the oscilloscopes. For differential probes, the useful voltage range is limited and they are more expensive than passive probes.
Probes must be compensated, since capacitances in the cable and in the oscilloscope can vary. For this purpose probes have a trim capacitor, mostly at the tip or in the BNC connector. One measures a recangular voltage provided by the oscilloscope with the probe. The knob of the trim capacitor is rotated until the signal becomes perfectly rectangular. Thereby it is ensured that the frequency response of the probe is correct and signals under investigation are undistorted. On this subpage there are further instructions, explanations and simulations on how to do probe compensation.