Modern Digital Voltmeters
Often referred to as a DVM, are usually based on an analog to digital converter (ADC) with about +/- 200mV full scale. The ADC would have a very high input resistance. Sometimes about 10 megohms although it may be essentially infinite in many designs.
They often have a low voltage scale where the terminals connect directly to the ADC. The input resistance is then either extremely high (1 Giga ohm or higher) or 10 megohm.
For other voltages there will be resistor dividers to scale the basic ADC input voltage to the required range. For example a 50:1 divider would give a +/- 10V range. It is common that the input resistance seen by the terminals is maintained at 10 megohms regardless of the voltage scale.
For measuring current a shunt resistor would be used to define the current range. For example a one ohm shunt resistor would give a 200mA max current. Various resistors would be used to define the current range.
At the voltages and currents the resistance of any switches in the path can seriously impact the accuracy so they are designed such that the contact resistance does not significantly affect the reading.
Depending upon the specific arrangement for current measurement the voltage drop of the meter can be more than the basic 200mV scale of the ADC. Maybe up to 500mV or so. In addition there can be significant voltage drop in the probe leads so you need to be careful when doing current measurements of low voltage circuits that the meter does not disturb the circuit you are measuring.
Other scales such as resistance, AC measurements or temperature can easily be provided by using conditioning circuitry to create a voltage that is proportional to the measured quantity and feeding it into the ADC.
Older Analogue Multimeters
These are usually based on a moving coil movement with a basic sensitivity of about 50uA, that is current rather than voltage although it will have several hundred to a couple of thousand ohms of resistance so will drop 50-100mV at full scale.
By putting resistors in series the voltage range which is measured can be defined. For example a 200 kilohm resistor would set the full scale to 10 Volts.
The basic current range would be 50uA but by putting low value resistors in parallel (a shunt resistor) the current can be increased to any desired value. Most of the measured current will pass through the shunt resistor with a very small proportion through the meter movement.
Care is needed with temperature variations as the copper coil in the movement will have a positive temperate coefficient and cause inaccuracies when paralleled with a shunt resistor.
Typical Analog Multimeter Circuit.
Image credit: Answer by devnull to the Electrical Engineering Stack Exchange question: What does the multimeter dial do internally?