Is my impression htat you can do the covariant derivative of a tensor field, but is possible also to do the covariant derivative of a single tensor? There are a plenty of webpages that says: "the covariant derivative of a tensor is this:" and then they give you a formula where appears the connection coefficient, but I do not know if this is correct

What do you mean by "a single tensor"? All tensors are tensor fields in that the tensor is defined throughout the space on which it is defined. Even something as simple as the position vector or the velocity vector are tensor fields. The covariant derivative applies to all tensors and vectors.

Is my opinion that a tensor field is a field with a tensor attached to each point. Just like a vector field is a field with a vector attached to each point
But I do not think that a vector and a vector field are the same thing

no, you can't take a covariant derivative of a single tensor, just like you cannot take a regular derivative of a single number. you take derivatives of functions, defined over smooth manifolds, and you take covariant derivatives of tensor fields, defined over (at least) an open set on a smooth manifold.

if you see any literature referring to the covariant derivative of a tensor, they are just being sloppy about the difference between a tensor and a tensor field

Actually I think you're right. A tensor is defined at a point P on a manifold according to certain criteria (e.g. how the components transform under a coordinate transformation). I made the mistake of assuming there are tensors defined at all points on the manifold.

Muchus Gracias!

Great! What about the Lie derivative? Do you take it of a tensor, or do you take it of a tensor field (or of both)?

I have a doubt: you have the stress-energy tensor of General Relativity, and then I have learned that the covariant derivative of the stress-energy tensor is zero. Does it mean that the stress-energy tensor is not a tensor, but rather a tensor field, right?