We take writing for granted. Starting in kindergarten, children are taught to write the letters of the alphabet, including their own names. In the Middle Ages, however, writing was a relatively rare skill, rarer for example than reading. For us, reading and writing go together, but if you think about it there's no necessary reason that they should.
Charlemagne famously was well educated, able to speak both Old French and Old German fluently, able to read Latin and, he claimed, at least a little Greek, but he couldn't write. He had never developed that fine motor control of his fingers (why we start kids in kindergarten), and a lifetime with a sword or a horse's reins in his hands had further coarsened them. He used to keep a wax tablet and stylus by his bed and practice if he woke up in the night.
Writing was rarer than reading because it was highly technical. Until the late Middle Ages, any permanent writing was done on parchment, sheepskin carefully prepared, which of course was far more expensive than paper. This is why rough drafts and quick notes were done on a wax tablet, that could easily be wiped clean and reused.
Writing was done with a quill pen--which in fact continued to be the case until the nineteenth century. So you needed a goose to produce the feather to use as a pen. (Our word indeed comes from the Latin penna, meaning feather.) If you were right-handed, you needed a feather from the goose's left wing, so it would curve away from your face as you wrote. The right wing feathers were understandably cheaper. This was not quite as big a deal as you might suppose, however, because the feather would be cut down to maybe eight inches long before use (not the enormous feathery pens you may see in movies).
A feather, being hollow, will draw up ink, but the scribe still needed frequent dipping. As the scribe wrote, the quill would wear down, so it constantly needed trimming with a pen knife. The knife was also used to split the quill, forming the nib, and to erase mistakes. Without modern erasers (or the backspace key), medieval scribes had to carefully scrape incorrect words off the parchment. Depictions of scribes at work, generally writing on a slanted lectern (as in the image above), often showed them with a quill pen in their right hand and a pen knife in the left.
The ink itself was usually made of soot, lampblack or charcoal, mixed with a binder. The sap of plum or cherry trees was considered to make a good binder. Some advocated boiling up hawthorne branches to make a thick, dark ink. Whatever the ink was made from, it would have to be thinned before use, generally with vinegar (that wine that went bad still had a use!). The prepared ink would be put in a horn for use (in images it appears to be the tip of a cow's horn). Most "black" ink was actually dark brown, although Italian scribes prided themselves on really black ink.
Charters would be written in black (or brown) ink, but books usually had rubrics, that is red initials and/or headers to individual sections. Someone copying a book would thus need to have both red ink and black ink handy.
Although we think of handwriting as very personal, in the Middle Ages different scribes at the same place were expected to write a very similar hand (although there was still some variation). One can indeed give documents a place and rough date just by the style of the writing.