In the context of image processing (and computer vision), the concept of convolution comes up a lot. Convolution is quite related to the concept of Fourier transform and DFT. In the context of image processing, we are mainly interested in DFTs, given that signals (images) are discrete and finite. In this context, I have heard the expression "periodic signal", which reminds me of a sine or a cosine waves, which have a period of $2\pi$, that is, every $2\pi$ the output of these functions repeats. Now, in the case of images, it is not clear to me the meaning of a periodic signal. What would be a periodic image? How is this concept related to DFT and convolution?
You can think of the periodic signals like sine and cosine are basis for building other signal. See this figure for a 1-D example.
For images, sine and cosine extend to 2-D patterns as something like in here. As in 1-D signal, any images can be represented as a weighted sum of these basis images. And roughly speaking, DFT are just weights in the weighted sum above.
Convolution connects with DFT by the nice convolution theorem. Namely, if you have two images (one of them is typically a filter, but you can consider any filter as an image patch as well) A & B, F(A * B)=F(A)F(B). That is, the Fourier transform of the convolved image of A and B is equal to the product of the Fourier transform of image A and the Fourier transform of image B.