Physicist: A “dimension” is a direction. So, we’re in “3 dimensions” because there are only 3 directions, and every other direction is just a combination of those 3 (and their negatives). For example, by telling someone how far north, east, and up to go you can direct them anywhere on Earth. Time is also considered to be a dimension, but there are some subtleties involved with that.
When string theorists talk about space having 10 dimensions they’re talking about all but four of those being “tiny loop dimensions”. We’re used to thinking of a direction/dimension as extending outward forever, but these tiny directions just make a short loop and bring you back to where you started.
For example, if the universe had two normal dimensions, and the third dimension was a “1-meter loop dimension” then you could move, say, north/south and up/down no problem, but if you looked east or west you’d see the back of your head a meter away. But unlike a mirror, you could physically poke the back of your head. Or give yourself a haircut, because why not.
The smaller dimensions are like that, but loop on a distance that is much, much smaller than the smallest particle. As a result, it doesn’t really make sense to talk about movement in these tiny dimensions, basically because there’s no place to go.
So, if you want to avoid being hit by something, you just step to the side (in any of our 3 dimensions). You have the option because you have the room. In the loop dimensions there isn’t enough room, so as far as we (and even particles) are concerned, they may as well not be there.
String theory (which is where all this “10 dimension” talk is coming from) is the only theory of everything on the table today, but unfortunately it hasn’t made any verifiable predictions (yet). The extra, but undetectable, dimensions are just one example of that.
The portal picture is from here, and the alleyway photo is from here.