Multimeters are electronic measurement devices which nowadays can at least measure current, voltage and resistance.
What features are necessary?
There are many cheap Multimeters (less than 25 Euros) - differently equipped with features. For many purposes a device for less than 10 Euros can be sufficient.
acoustic conductivity test
One of the most important features is an acoustic feedback for conductivity. Whenever the probes of the Multimeter are connected to two poles of a conductor (like 2 ends of a wire) it beeps. This is an important feature for trouble shooting in circuits and thus it is a must have.
on/off selctor and auto-off
Most very cheap multimeters don't have an on/off switch but only a rotary selector which has one position for "off" and all other positions mean the meter is turned on and meausuring - draining power from the battery when forgotten. Better multimeters have at least a dedicated switch for on/off which saves all the work like "rotate to off when work is done" and "rotate back to voltage measurement when you need it again". While this is more convenient it still doesn't help if you forget to turn the meter off after work. The most convenient type of on/off mechanisms is an "auto off" feature. After a certain amount of time the meter simply turns off. Some meters also have a warning beep before the shut off. (and even better units have a switch which disables the auto off for measurements which have to run over hours).
Auto range
A convenient, but not necessary feature is the "auto range". On many multimeters you have to select a range, for example of voltage to measure. if the meter is set to 10V it can measure voltages from 0 to 10V. Higher voltages cause an error to be displayed on digital meters and on analog meters the needle goes all the way up to the end of the scale. Similar for 100V setting the meter can measure from 0 to 100V - however with less accuracy for low voltages.
Capacitance
Most cheap multimeters don't have a selection to measure capacitance. If you're not designing or modifying circuits this will probably be no necessary feature. For example an electrician doesn't need a multimeter with capacitance measurements in almost 100% of the cases as it is no use when checking cables, connections, lamps, fuses, etc. However if you are often dealing with components which are unknown or hard to identify - the capacitance measurement may come handy to find out what part you are dealing with (for example there are capacitors looking similar to resistors - they can easily be identified as resistor or capacitor as resistors don't have a capacitance you can measure with the multimeter). The same applies for inductance measurements - however the capacitance feature is more important than inductance - there are also fewer meters with inductance measurement.
Diode check
Many Multimeters have a combined measurement for connectivity and diode check. When connected in forward direction current flows through the diode and a voltage (the forward voltage) is displayed. When connected in reverse direction the diode doesn't conduct which is indicated by the meter as "not conducting" (the same is displayed when nothing is connected to the probes). The diode check is also a useful feature to check the function of an LED. When connected in the forward direction a bright LED emits a tiny amount of light. While for red LEDs a voltage below 2V is sufficient, blue and white LEDs need above 3V to be tested. Many diode testers don't use more than 3V - so the meter will display the LED as not conducting in both directions and the LED will also not light up. Unfortunately this is often not indicated in the description of a multimeter - so you'd have to try that on your own.
Conclusion - what will I buy?
If you're unsure if you will need all the pro-features, simply buy a multimeter which is cheap and has an acoustic conductivity test. If then you're later deciding to have fun with electronics, buy a meter which 100% fits your needs. You can then still use the cheap meter to take with you - or keep it as backup in case the batteries of your the good one are empty. Also it's always good to have 2 multimeters, for example to measure current and voltage simultaneously - which can be done with cheap meters without any issues.