When I first started playing the guitar, I started out on an old beaten up acoustic guitar handed down to me from my grandfather. The action was so high that it was quite tough to play, but I was determined as ever to be able to play some of my favourite songs. After a couple of months of having a hard time making that guitar sound the way I wanted it to, I was lucky enough to get an electric guitar for my birthday -- and this changed things in some great ways. With the electric guitar it was much easier to play and learn (lower action), and I was able to somewhat recreate the sounds that I was listening to on my records (ahem, CD's) at the time -- although my tone just wasn't fully there. After playing with a really cheap inexpensive amp for some time, I was able to purchase an old Peavey amp from a friends father -- and this made a HUGE difference in the sound I was able to get from my electric guitar. This set me off on a path to really get into exploring the sounds of different guitars and amplifiers over the years, and I can tell you that an amplifier does almost as much (if not more) for your sound as your actual guitar.
I've owned many amplifiers over the years - starting with the old Peavey amp (I can't even remember the model, but it was from the 70's or 8o's), to a more modern 90's Peavey Bandit 112, to a Fender Twin, Peavey JSX, Krank Chadwick (like a Marshall JCM800 clone), Marshall JVM100 and a TON of practice amps, both solid state and digital. Nowadays I mainly get my recorded tones from my Line 6 Helix, which I think of as my swiss army knife for guitar tones. I bring all of this up so that you know I've been through many kinds of amplifiers, and know the benefits of each for specific situations.
These days there are so many options when it comes to buying an amplifier - so I'm not going to talk about specific brands. But I will talk about what kind of amplifier might work best for you, so you can at least head in the right direction.
There are 4 main types of amplifiers that we'll talk about:
Solid State Amplifiers
The last 2 are very similar, but I'll explain why I created two separate categories in when we get there.
Within these 4 types of amplifiers, there are also different power and size options - so that will weigh into what amplifier will work best for you as well. A large amp typically needs to be played at a louder volume to make it comes alive, so having an amp too big for your situation should be avoided if possible.
A tube amplifier (or valve amplifier) uses pre amp and power amp tubes to increase the amplitude or volume of the signal. These provide a very classic guitar sound, and many guitarists prefer them because of their raw power and tone. They can get LOUD! Many enjoy the warm tone of a tube amp, but the downsides would be the initial cost (they can be quite expensive) and the fact that the tubes need maintenance (they need to be replaced every so often, just like a lightbulb). Tube amps typically needs to be turned up loud to get that nice warm sound out of them. Because they are so loud, a tube amp isn't always favoured for practicing -- although nowadays you can get tube amps in small packages with very low wattages that can be played at 'bedroom volume' without sacrificing tone. Some examples of tube amplifiers would be a Fender Twin, Marshall JCM800, Vox AC15 etc.
A solid state amplifier is an analog amplifier (like a tube amp) that uses diodes and transistors (instead of tubes) to amplify the signal. The tone can be considered more 'sterile' but the one thing these amplifiers excel at is producing super clean sounds at loud volumes. You don't see as many artists using solid state amplifiers in a live setting as you do tube amplifiers, but that doesn't mean they don't have their merit. The initial investment is less since they don't cost as much as a tube amplifier, and they typically require less maintenance (no tubes to change). Generally speaking, solid state amplifiers are not as loud as a tube amplifier, and they don't have the nice smooth natural distortion that tube amplifiers have. But that being said, a solid state amplifier can sound great if paired with the right guitar and pedals. Some examples of a solid state amplifier would be a Roland JC-120, an Orange Crush, and too many bass amps to name.
A digital amplifier is essentially a solid state power amp, with a digital pre-amp. What this means is that you can add in a whole ton of effects and sounds you wouldn't normally be able to achieve with just solid state circuits. Digital amplifiers typically have different amp models and effects to choose from, and most allow you to save presets so that you can easily revisit them later without having to memorize the settings. These are getting quite popular these days, as you can experiment and achieve tones that you would never be able to get without buying a ton of pedals and different amplifiers. Examples of these would be the Spark Amp, the Fender Mustang Amp, and the Boss Katana.
A digital modeller is just like a digital amplifier, but without the power section. These can be standalone units (like a pedal or head), or even a program or plugin on the computer. I'm putting these in a different category because they serve a different purpose - these are more for a studio musician, or someone who performs live and wants to be able to pair it with their choice of power amp (whether it be the PA system, a stage monitor, or even run it through the power section of another amplifier). The benefits of these are the same as a digital amplifier, but it allows you to pair it with whatever power amplifier source you would like (even if that just means listening through your studio monitors or computer speakers). The higher end units in this section can actually be quite convincing when it comes to recreating the sound of a solid state or tube amplifier. This has personally become my favourite type of amp to work with, but it does require a large investment, and also can take quite a bit of time to learn how to use properly. Examples of these would the the Fractal Axe FX, Line 6 Helix, and the HeadrushFX. (***You can also get a digital amp simulator on your computer -- examples of this would be the Helix Native, and the Neural DSP***).
It might seem like a lot of information to look into, and there certainly are a ton of amplifiers on the market. I'll try to provide some scenarios and explain which amplifier option is best for you and why. These aren't clear cut as everyone is different - but at least it will give you some direction
You're a beginner guitarist with no knowledge of what you like or don't like: I would suggest a small solid state or digital amplifier for practice. A digital amplifier is a good choice here, as it will help you learn about different tones and what you like and don't like, with a smaller initial investment. A solid state amplifier might be a better choice if you think you'd like to buy pedals (overdrive, delay, reverb etc) in the future.
You want an amplifier to practice with at home, but also be able to jam with others: this time I would suggest a larger sized solid state or digital amplifier, with the decision based on whether or not you have pedals or would like to buy pedals in the future. You could also look into a smaller sized tube amplifier that will allow you to practice quietly if needed.
You want an amplifier mainly for jamming with others and performing, and would like to put pedals in front: for this scenario I would recommend a tube amp, as you will have no problem with volume here. Depending on how loud you need to be, you can easily get away with a smaller tube amp, since they tend to be much louder than their solid state and digital cousins.
You want an amp to practice with at home, with the option to record: I would suggest a digital amplifier, since most of these can double as a guitar interface and allow you to output your tone straight to your computer without the need for a microphone (or to be loud).
You want to be able to record a variety of tones through your computer: for this I would recommended a digital modeller or amp sim, since your main use will be for recording. Over the past few years these have really started to sound like the amplifiers they emulate. You can simply use your studio monitors or headphones to hear your tone (no need to have a power amp/speaker in this situation.
Of course there are many more scenarios, but hopefully this gives you an idea of where to start. Even if you've narrowed down which type of amplifier is best for you, there is still the task of researching all of the options in that category and figuring out which one works best for you (and believe me, there are a lot of options out there).