Anastasiya has some great info! I've answered briefly via video here, but to elaborate a bit:
If you are a beginner, the most important thing is that the instrument is in good, working order. There are lots of "bells and whistles" you can get when you purchase a new instrument, but at this stage you aren't going to know what to do with them. I usually recommend that my students wait on the expensive stuff until they are ready to commit to an intermediate level instrument.
Beginner flutes are usually made of nickel with silver plating, have closed holes, a C foot, and an offset G key. The closed holes and offset G make things a little easier when you are starting out. The C foot will work great for a few years until you start playing more advanced pieces, and you won't need to worry about materials until you upgrade to your next instrument. There are many brands to choose from; Gemeinhardt, Yamaha, Armstrong, Jupiter, and Pearl, are all brands I've either played myself or have encountered with my students that work great, but there are also many other brands out there that work perfectly well for beginners. You don't need top of the line, and used flutes are great, too! Just make sure you take used instruments in to a flute repair technician to make sure everything is in working order. Most of my beginner students are able to find good, working instruments for $100-500.
When you upgrade to an intermediate flute, you'll want to consider several other factors. 1. Material. This is the most important, but also the most expensive factor. If possible, you want to get a solid silver body and head joint. If that isn't within your budget, at least get a solid silver head joint. Many people are tempted by engraved or gold lip plates on their head joints at the intermediate stage, and it does look very pretty, but it is unlikely that it will make much of a difference in your sound, and it will definitely be at an additional cost. 2. Open holes. You want to get an instrument with open holes. Open holed instruments usually come with plugs to fill in the holes while you are getting used to the holes, and your teacher can guide you through removing them. 3. B Foot. Not 100% necessary, but having the B Foot is very helpful (and fun). It allows you to play an additional low note, which you'll need for some pieces. 4. Other options that come down to personal preference. You can get a split E mechanism, which makes your high E (often a nasty note) a little easier, or you can get an in-line G, which changes your finger posture on your left hand. Most flute makers will make you choose one of these two options due to mechanical factors. Gemeinhardt and Yamaha are great brands in this range, but again, there are plenty of other brands that are also great. I'm usually able to guide my students to intermediate flutes in the $1000-2500 range.
I won't go into professional level instruments very deep here, because if you're looking for one, you already know what you're looking for, but for those who are curious, professional instruments are generally at least solid silver, often with additional metals like gold or rose gold, they generally have a B foot and open holes, and the other specifications to the player's preference. They range roughly $7000-13000 most of the time, but can go much, much higher.
I hope this is helpful, and good luck!
