How can extra dimensions be small?
Small means that the extra dimensions form a compat space (you can think of it as a space with finite volume). An example would be the 4 dimensional space that we know and love (3 space + 1 time) plus an additional fifth dimension as a circle. This is denoted as $$ \mathbb{R}^{3,1} \times S^1\,. $$ There are 4 coordinates $x^{\mu=0,1,2,3}$ plus an extra periodioc coordinate $\theta \sim \theta + 2\pi R$. This is of course just an example. The constructions arising from String Theory are way more complicated.
Now the concept of big and small in this simple example refers to the size of $R$ in units of the typical length scales of your experiments. If $R \ll \hbar c / E$ (where $E$ is the typical energy of your favorite particle collider), then no experiment will ever be able to resolve that circle.
Another way to look at it is the following. Since the extra space is compact we can expand our fields in a Fourier series (I don't know your background, so what follows might be incomprehensible)
$$ \phi(x,\theta) = \sum_{n=-\infty}^\infty e^{i n \theta/ R}\, \phi_n(x)\,, \qquad \phi_{-n} \equiv \phi_n^\dagger\,. $$
And the kinetic term in the Lagrangian becomes $$ \frac12\int_{\mathbb{R}^{3,1}} \mathrm{d}^4x\int_0^{2\pi R}\mathrm{d}\theta\,\big(\partial_\mu \phi \partial^\mu \phi + (\partial_\theta \phi )^2\big) = 2\pi R\sum_{n=0}^\infty\int_{\mathbb{R}^{3,1}}\mathrm{d}^4 x \left(\partial_\mu \phi_n^\dagger \partial^\mu \phi_n + \frac{n^2}{R^2}|\phi_n|^2\right)\,. $$ So now we are back to 4 dimensions, but we have a whole new tower of massive particles instead of just one. These are called Kaluza Klein modes. Again you see that we will never be able to probe the existence of such particles if their mass (which is proportional to their inverse radius) is much bigger than the energy of our bigger collider.
In the more general case instead of a circle you'll have a manifold with possibly more than one dimension, but the logic is the same.
Think about it this way. Suppose there's an ant crawling on a very thin, stretched wire located 100 meters from you. How would you describe the ant? You would be very much justified to say it's crawling on a 1-dimensional line and use only one coordinate to describe its position.
However, if you zoomed in, you'd see that the wire is not a 1D line - it also has a radius. That means that, strictly speaking, you need the full three coordinates x,y,z to describe the ant's motion. Two of the coordinates would barely vary regardless of where the ant is on the wire, however, a sign that they are "small".
Extrapolating from this to extra dimensions, being small allows extra dimensions to hide. If the extra dimensions are big, then we would have seen them already.
In string theory, where extra dimensions most commonly show up, these extra dimensions are in a fundamental sense really equivalent to the 3 familiar spatial dimensions: a priori, they play the same role and obey the same laws as the 'large' dimensions.
However, string theory allows one to play around with what happens to these dimensions. Although all 9 spatial dimensions are equivalent from the start, there can be solutions of the theory in which something has 'happened' to some of the dimensions. Commonly, some dimensions (usually 6 of the 9 spatial dimensions) are taken to be compact. For instance, some of them can go periodic: walking in a straight line in such a dimension will at some point bring you back to where you started. Such compact dimensions have length scales associated to them, and if that length scale is very small, the associated dimensions are not directly visible anymore to processes happening on larger length scales. For a periodic dimension, the length scale would be the distance one had to walk in order to return to one's original position. If this length is as small as the size of a proton, the periodic dimension would disappear from everyday view.
These dimensions do, however, leave an imprint on the effective lower-dimensional physics. The compact dimensions can contain a bunch of D-branes, they can have various topologies and so forth; all these influence the physics in the effectively (3+1)-dimensional world.