Electric guitar - buzzing when playing a string?

I have recently purchased an electric guitar and have been learning for the past 2-3 days. I've gotten to the point where I can play the intro to the rock n roll riff, but when I press a string (mainly the thicker ones such as 6 and 5), I hear a buzzing sound like the string is slightly touching the next fret over. What could the reason be? Am I pressing the string incorrectly, am I pressing too hard or too soft (mainly happens when I use my pinky finger, but has happened with other fingers also), are the strings too high or too low, or could it be that it is out of tune (I'm pretty tone deaf, so I think the tuning might be far from perfect)? Please let me know if any of these are possible reasons or there is another reason why it might be doing this. I just want to add that when I press string 4 it sounds awesome, I get a nice clean tone, but with the others I get a tone and a buzz. Thank you!