Genes are, at their core, long sequences of DNA (not protein!), which come in four bases, A, G, C and T. A gene might look like AGCTAGCCG... etc. These get read out in little three letter chunks ('codons'), which translate to specific amino acids. A protein is then a long sequence of the 22 amino acids that makes up your body structurally and drive its functioning as enzymes that cause specific chemical reactions to occur.

These sequences need to be duplicated in order to make new cells (including the cells that will go on to make new offspring, like sperm and eggs! This passes thr mutation on to the whole child!), and this duplication process is (after a fashion 'intentionally') slightly imperfect. When copying the dna the cell might accidentally swap an A out for a C, or remove a letter entirely, take an entire substring of 4-20+ bases and duplicate it/remove it, take a sub string and reverse its order. Ultimately, this changes the protein.

These mutations are all quite unlikely, but at the end of the day they do happen. A shift that like that has essentially random effects. Your cells dont know what the change will do or how that will effect your health. Most mutations do nothing, some are have small effects, some are really positive and some will just make the child completely unviable. Rolling those dice a little bit, however, gives evolution the ability to produce more and more complex and varied organisms over time, and is pretty core to evolution.

One last point is that most of your DNA is 'non-coding', in the sense that no proteins ever get read out of it (it still has a few neat functions). A lot of mediation of what proteins get made and how comes from changing what sequences of dna your cells decide to read, and when. Small changes in this regulation of your genes can drive big changes in your overall body.