There are many different types of microphone, all with specific purposes in mind, but for most audio and performance use it will come down to the two most popular designs, which are Dynamic and Condenser.
The difference ? Huge subject ! but the main technical difference is that a condenser mic requires power to operate and a standard dynamic does not.
A condenser mic uses a form of charged capacitor, with changes in air pressure over the plates causing variations in the capacitance level which is converted into a small electrical signal, which is then fed to a preamp. A Dynamic mic is a lot simpler, and uses a diaphragm and electromagnet to create its signal, it is basically the reverse of a speaker, with your voice moving the cone and a current being created by the coil from that movement. Condenser microphones generally have a much greater frequency response, and transient response, as they react quicker to changes in pressure and frequency. They also have a much louder output, but are more sensitive to loud sounds and are generally more fragile units due to their complex internals. They come in a huge range of sizes, from the tiny units in mobile phones, right up to ceiling mics for halls which can be the size of a 2 litre drinks bottle. Used in recording studios, auditoriums, halls and churches, they are exceptional at picking up the wide spread of sound from a choir or orchestra, and there is no real dynamic equivalent available for those uses.
Dynamic mics are considered much more robust, and specific design variations mean their directional sound pickup can be altered and their frequency response can be changed by using larger or smaller diaphragms. They are the industry standard for stage singers, drum mics, guitar cab and instrument mics, and for speeches and other general situations. This is down to a few factors, with reliability and durability being high on the list, and the fact they don't require a power source, but the main feature is their polar pattern, or the direction to which the mic is sensitive.
Many lower cost dynamics have a full pickup pattern, which is known as uni or omni directional, and will collect sound from all directions. This is fine for many applications, a quick Dj announcement or bit of karaoke, but will be picking up everything in the room at the same time. A professional vocal dynamic, such as the legendary Shure SM58, uses what's known as a cardioid pattern (heart shaped) which is directional, so will only be picking up from the front and blocking the rest out, and is a favourite for live performers as it rejects feedback and is very consistent in signal level.
Historically, condensers have always been more expensive to buy, as they are a fair bit more complex internally, but far east manufacture has dropped this level significantly, and there are now units available for very low cost if you really want to dip your feet into that technology, though your main consideration of any mic purchase must always be intended usage. If it's going to be chucked about, rained on, thrown in a bag with cables at your gig etc, then its dynamic all the way as they can take that sort of abuse and still live a long life.
If however, you're putting a studio together, recording vocals or instruments in a controlled environment, or need small lapel mics or a large ceiling unit, the condenser will be the better choice.
So as a buyer, you must consider what you want from a microphone and where it's going to be used. It's one of the few products where cost is irrelevant in comparison to choosing the correct type. You can spend £3000 on a beautiful studio condenser unit, but its inherent qualities will become major flaws if you try and use it for your heavy metal band or outside in the wind, and you would get better performance from a 30 quid dynamic. This obviously works the other way around too, so intended usage must always be your first and overriding decision.
Quality wise, like anything electrical, spending a little more will get you better built units, with better internals and a longer lifespan, and obviously there are known brands that can give you that reassurance of their proven track record, but the budget market is very capable now too and shouldn't be overlooked, as they are perfectly fine for general use.