"The other café" is a fine way to refer to a single café that is different from a café you were speaking about earlier. "Other" contrasts this one, specific café from an earlier café. "Other cafés" is what you would use if you had previously spoken about a café, and now you want to speak about numerous cafés which are different from that first café. "Another café" is again speaking about a singular café, and contrasts it with an earlier café you were speaking about. When you use "another," you are free to contrast more cafes later on. It is like a chain. I like the McDonalds café, but another café by the French Villa is also nice to sit outside on a summer afternoon, then there is another café which I find expensive by the church on the corner, etc. "The another café" is incorrect, and would never be used. It combines "The" which is the definite article, with the indefinite article implied by "an-other" so it is like saying 'The an <something>" which is always going to be wrong.