Every ham should have a multimeter and know how to use it. Multimeter use is a practical skill not only for ham radio but also around the house for general power and wiring work. As with many topics on this site, details of the subject are too extensive to cover in a simple post so we will give you just basic info along with some resources for further study on your own. Even with minimal detail this is still a long, involved post.
By definition multimeters measure more than one thing. In electrical work a multimeter typically measures voltage, current, and resistance. Sometimes it is called a volt-ohm-milliammeter or simply a volt ohm meter (VOM). Note that these are the three fundamental electrical parameters as described by Ohm’s law.
In addition to measuring the three core electrical parameters, multimeters may also read other things such as temperature, frequency, capacitance, plus provide quick checks of diodes and continuity. At minimum they will measure voltage and resistance, since these are the two most commonly read values.
To measure voltage you connect the meter leads between (across) two points. This parallel connection allows real-time undisturbed readings in live circuits or power sources.
Note that resistance measurements are also made with test lead parallel connections but never on live circuits; more on that below.
Measuring current with a common multimeter is more disruptive because it requires the circuit to be broken somewhere and have the meter leads inserted in-line to read amperage, and that is not often very convenient.
This series connection requirement for measuring current is the main reason it is less commonly used, and why clamp-on (non-contact) current meters are sometimes favored at the expense of accuracy and greater cost.
So what would the ham or handy homeowner need a multimeter for? The possibilities are endless but common scenarios are testing batteries, locating blown fuses, verifying DC and AC power supply voltages, and checking cables for undesired opens or shorts. For kit or DIY circuit builds it is also useful to verify resistor values and test active circuit voltages. Here is a link to a list of Ten uses for a multimeter from Ham Radio School.
There are a few safety considerations to note when using a multimeter. Two involve the test leads which connect the meter terminals to the circuit being measured. First, the leads (probes, wires, and connectors) should be rated for at least the voltage being measured. The meter itself must also be rated to exceed this voltage.
Good meter leads will have shrouded plugs, high quality flexible insulation, and finger guards (flanges) at the probe tips:
They will also be rated at least 600V as marked on the probe for general measurement below that voltage. Special high-voltage probes are used for measuring higher voltages, but it’s unlikely that the average ham will need to do so.
You may come across some old and/or cheap test leads which have no rating, no shrouded plugs, and no finger safety guards. These should be avoided due to increased risk of shock when measuring hazardous voltages.
The second safety consideration involving test leads is to not blindly trust them. Unreliable connections within the test probe set frequently occur. So if you are using a meter to verify that no hazardous voltage exists, it will read 0V with defective (open) test probe. Best practice is to test the meter and probe on a known voltage to make sure it indicates correctly immediately before doing a hazardous voltage safety check. At least perform a quick continuity verification of the leads by touching the probe tips together on resistance setting before switching the meter back to voltage.
This suggestion highlights another safety consideration when using a multimeter: that is to ensure that you do not measure live voltage with the meter in resistance (ohms) setting.
In resistance mode the meter is looking for a low-level signal and has its internal current source actively driving an output. Should the meter leads be connected to an external voltage source in this situation, damage to the meter is likely to occur, and possibly a violent eruption inside the unit (if applied voltage is high enough) which might cause fire or operator injury.
When measuring a hazardous voltage (more than 24V, generally) take great care with your hands to ensure no incidental contact with live conductors (avoid shock). Use only one hand where possible.
Multimeter appearance, features, and cost vary greatly between manufacturer and model. In addition to what parameters are measured, some will have a fixed scale, meaning the operator has to select the expected input range, vs auto-ranging units which scale the measured values based on measured input. In both cases you will always have to select the basic functions of AC voltage, DC voltage, resistance, and current (AC or DC). More expensive models will read non-sinusoidal AC values known as true root mean square (TRMS) for special applications, The average ham or DIY homeowner is unlikely to need TRMS.
Two basic types of multimeters are used in amateur radio work: Analog and digital.
Old-school analog meters feature a needle which moves over a reference scale and require visual interpretation of the value. These work well for a few applications such as when the signal being monitored is changing, as the moving needle highlights the change. 
Digital Multi Meters (DMM) feature a numeric readout of the actual value. The combination of direct readout and good interface circuitry give digital meters a distinct advantage over analog for general use.
Digital multimeters by design have high input impedance. Analog multimeters may also have an active interface circuit on their input, in which case it is termed a vacuum tube volt meter (VTVM), or electronic equivalent. In either case the intention is to minimize influence of the meter on an active circuit (load). 
Buying a multimeter for home and/or amateur radio use is not difficult or terribly expensive. You choose which features you want and shop what is available. Avoid the super small and cheap meters which tend to compromise on safety and reliability. A good basic DMM should cost in the $50 to $200 price range.
Technical Note
A primary reason that multimeters are relatively affordable while supporting different parameters is that at its core, nearly all readings are basically displaying DC voltage:
- DC voltage is scaled to the core range by using resistive voltage dividers
- AC voltage rectifies to DC then scales to core DC range
- Current is measured by reading voltage across a precision shunt resistor, scaled to core DC range
- Resistance is measured by reading voltage across the unknown external resistance driven, by a precision constant-current source, scaled to core DC range
- Frequency is derived from a simple F-V converter circuit, scaled to core DC range
- Temperature is derived from voltage across thermocouple or RTD elements, linearized and scaled to core DC range
- Capacitance is measured by voltage after charging the unknown value with a constant-current source, scaled to core DC range
In essence, a multimeter is really just a narrow-range DC volt meter with supporting circuitry to adapt and interface to different desired parameters. Mass-produced specialized multimeter integrated circuits put much of that on one small chip. A rather intriguing bit of technology.
Useful web links
How to use a multimeter for beginners (YouTube video)
How to use a digital multimeter
Basic electronics: The Multimeter– ham perspective
Every Amateur Radio operator needs a multimeter. Or two. Or a bunch… lots of good info
All about multimeters– ham perspective (YouTube video)
How to take basic multimeter readings– ham perspective (YouTube video)
How to use a multimeter for beginners: Part 1 (YouTube video)
How to use a multimeter (YouTube video) ** best I have found to date**
NewHams.info 