How do current probes load a circuit?
I didn't know there were many types, just AC and AC+DC. Is that what you mean or is there another sort of classification?


There are several common techniques for measuring current. The standard terminology is to discuss the "burden" the meter places on the circuit and to minimise the loading effect the burden should be kept as low as possible.

Shunt resistor 



Figure 1. A 500 A - 100 mV shunt. Source: RAmmeter.


This is placed in series with the load being measured. 
It will cause a small voltage drop and this voltage can be measured and converted to current by Ohm's law. 
Shunts can be used to measure AC or DC.
Note the two large bolt holes for the high current connections and the two small screws for the voltage measurement connections to avoid measuring voltage drops across the current connection bolts.


Current transformer 



Figure 2. A current transformer with standard 5 A output. Source: TestGuy.


The primary of the CT is in series with the load being monitored and the secondary goes to a measuring circuit - in the old days, typically an AC meter. 
They only work on AC. 
The burden seen by the circuit is the secondary burden divided by the turns ratio squared.
Often the primary consists of only one turn.


Hall sensor



Figure 3. A crude Hall sensor. Source: Connecting Industry.


These are like a CT but contain a Hall sensor in the ferrite core and this measures the magnetic flux induced by the primary and so can measure AC and DC.


See also Rowgowski coils and PCB current measuring probes for more interesting reading.