Good question! I'm serious, it's not a dumb question. I get a lot of people that ask me this with an apologetic, gosh-you-must-think-I'm-an-idiot tone and I just have to repeat myself here so y'all in the back can hear, this isn't a dumb question and I appreciate you asking! When I started getting serious about lettering, I didn't know what it was called, much less the difference between all these other practices. I'd always called everything "drawing and playing around with letters."
Let me break it down for you:
Handwriting: Just your basic everyday grocery list, meeting notes, class notes, jot your phone number down for me writing. It's probably come a decent way since those first learn-your-letter sheets. Print or cursive or somewhere in between, it all falls under the umbrella of handwriting. Some people's handwriting is font-like. Some people's handwriting is...not. (Count me with the latter.)
Hand-lettering: Artistically rendering letters (glyphs) using a variety of shapes, patterns, shadows, alignment, etc. Basically, illustrating each letter or word. Often used in a form of lettering composition which, in short, is putting hand-lettered words together in a way that delivers a quote, saying, or whatever in an artistic manner. It's a broad spectrum. I love this example by Risa Rodil.
Calligraphy: Think Declaration of Independence and most of the fancy wedding envelopes you see online. Calligraphy is not cursive; the letters are formed and connected in a different way than simply writing in cursive and it's not always a script. Blackletter is a great and recognizable example of a form of calligraphy. Traditional calligraphy is very measured and has various forms--like for example, the famous Spencarian script. Modern calligraphy is a more playful take on the formality of traditional calligraphy.
Typography: The style and appearance of type. Creating and manipulating fonts would fall in this category. Hand-lettering technically doesn't, but a lot of the internet would argue with that so it kind of just gets swept in. It's a gray-ish area and this isn't an in-depth look at the nuances of dealing with letters.
Cayligraphy: I've been lettering through classes, sermons, meetings, and everyday life for as long as my mom can remember. My family jokingly started calling my lettering "CAYligraphy" -- a punny play
on my name and my passion. The name stuck and so did the passion. When it came time to name this, it seemed natural due to my love for puns, love for all things letters, and fated name of Caylie. I finally came to love my name, after all these years. (Sorry Mom & Dad). At the moment, Cayligraphy has come to mean lettering, calligraphy, and illustration across a variety of mediums. The fun part of running a business with a made up word is I can really make it mean whatever I want. I think regardless of what it means in the future, it will always have some form of lettering involved.
Now, you should have a better idea of what we're referring to when we talk about lettering/hand-lettering, calligraphy, typography. And a mini explanation of where I got my business name. If you still have questions, comment below and I'll do my best to answer or point you in the direction of someone who can.