Conferences are where academics go to present their work. Academics are publishing papers all the time of course-- that's the main means by which science is shared. However, scientists these days are also expected to share their work in person. So we gather in gigantic buildings and sit together to listen to talks and give poster presentations. In a poster presentation session, you can walk into a room filled with various people lined up in front of posters describing research they want to tell you about, and in talks people present their work anywhere from 10 minutes to an hour.
Conferences are a lot of fun because you have a huge gathering of people all interested in similar topics. You don't have to motivate your work at a conference as much as you do normally, because everyone baseline CARES about the ideas and will have a somewhat related scientific background.
Everyone should try to get to a conference if possible in undergrad. There are two reasons to go to a conference: 1) you're interested in the topic and want to hear what the experts in the field have to say, 2) you're involved in the field and want to get feedback on your work and hear about others' work. In the former, you don't have to prepare anything, you just have to show up. In the latter (the more prestigious) you are going to be presenting your work to an audience!
What are the ways that you can present work? A paper is a final product-- that's the end goal of all of your research. Before people submit papers, they'll often give talks or poster presentations, which contain the research that will go into papers in the future. In talks you're invited to speak for a certain amount of time, and you give a powerpoint presentation that has a nice story to it. (There are good and bad examples of talks. I might write about it someday, but I really hope that you have the opportunity to give research talks in undergrad-- try to seek them out if they're available.) Poster presentations are something very unique to STEM fields as far as I'm aware. You stand in front of a 4x6 ft poster (yes, this is the size it almost always should be. There are very stringent rules for what a poster should look like and what it should contain) for an hour or two, and wait for people to come up to you. Then, depending on the conference, you either give them a 1-minute, 5-minute, or 15-minute spiel about what you did. Poster presentations are great because you can interact with the speaker directly, and if you're at a conference I highly recommend going to see the posters because if you don't know anything, since you're talking close to one-on-one with the presenter, they have to EXPLAIN things to you :).
Posters need to have very specific content on them in very specific formats. You're balancing two competing interests-- the goal is to have only pictures on your poster (because that's the mark of a good presentation; they should be listening to you, not reading) but you also want to have enough content on your poster that if you step away for a bit, someone will still be able to understand what you did. Posters thus have wildly varying amounts of text on them, but it shouldn't have "too much" (whatever that means). Posters are also organized the way papers are: Abstract, Introduction, Methods, Results, Conclusions. These are arranged in specific ways across the page in three columns (always). Basically, there are a lot of rules for posters, and you should Google them online and see what other people do before trying to make your own. Also, I'm sure your lab has presented posters before, so take a look at theirs.
So how do you get to these "conferences"? Turns out there are a LOT of conferences on a lot of different topics, and most cost money. The hardest ones are the ones where you travel, so that you'll need to pay for plane tickets, hotel tickets, and registration fees. However, schools often have opportunities for their students to present at local poster sessions or mini-conferences, so keep an eye out for those. Moreoever, sometimes schools will have scholarships that will let you attend conferences for free, or will fund you if you submit something. The best way to take advantage of conferences is to submit something-- have done enough research that you can present work in a poster or a talk-- and there are actually a lot of undergrad conferences where you will have the opportunity to do so. You're likely not going to have enough research to present at a major conference (such as the Society for Neuroscience conference, which assembles something like 30,000 neuroscientists in one place every year) but definitely look for the smaller conferences (usually a hundred or so people) especially the undergraduate ones.
I want to reiterate one more time that it is important to attend conferences. It's important to get the scope of science being done outside your local environment, and to see how scientists interact with each other. It's important to get experience presenting your work so that you're able to communicate your science orally, which is an essential skill in today's scientific world. It's important to meet other people and see what others are doing, and have practice in asking about other people's work and how to ask questions. It's important to learn how to present information in a specific poster or talk format. Conferences are how the scientific community communicate with each other, and so you want to attend those meetings as soon as possible :).
All right, last points. How do you figure out what conferences you want to attend? This is a hard question. I've done it historically by signing up for my school's talk / poster sessions, presenting at the mandatory poster sessions at the end of summer research programs, and by Googling nearby undergrad conferences if I'm looking specifically to present. For the bigger conferences, ask where your lab goes, because those will be the most important ones in your field. For neuroscience, and specifically computational neuroscience and vision science, most of my friends go to SfN (Society for Neuroscience), VSS (Vision Sciences Society), and CoSyNe (Computational Systems Neuroscience). For cognitive science the main conference is CogSci (Cognitive Sciences Society). I've just been advised that for machine learning, the main conferences are NIPS and ICML, for deep learning attend ICLR, for robotics attend ICRA, for computer vision attend EECV and CVPR, and for natural language processing attend ACL and EMNLP. (We do like our acronyms :)). These are national or international conferences with the largest gatherings of scientists or computer scientists, and then there are smaller, more specific conferences you can also attend once you focus your interests.
In short, conferences are how the scientific community interacts outside of papers and are very important in the community, and you should try to attend one (or present at one) and enjoy them!