What do the indices in 4-vector notation indicate?

The confusion stems from the fact that one commonly wants to consider what are called passive transformations as opposed to active transformations. The idea can be seen in three dimensions and then generalized from there.

One should not write ${bf u} = u^a$, it is an abuse of notation. But one can use a symbol without indices, such $bf u$ or $U$, in more than one way, as I shall explain.

Suppose $bf u$ is a vector in three dimensions. Then we can write $$ {bf u} = u^1 hat{bf e}_1 + u^2 hat{bf e}_2 +u^3 hat{bf e}_3 $$ where $u^i$ are the components of the vector and $hat{bf e}_i$ are some set of basis vectors. Let's suppose those basis vectors are aligned along the coordinate axes of some rectangular system of coordinates.

Now consider the effect of a rotation $R$. We can rotate the vector ${bf u}$ so as to get some other vector ${bf v} = R {bf u}$. This is called an active transformation. The vector changes to a different vector.

But we can also consider a rotation of the coordinate axes, without rotating our vector $bf u$. This is called a passive transformation because $bf u$ does not change. If we rotate the coordinate axes then we will find the vectors along the new coordinate axis directions are not the same as the ones along the old coordinate axis directions, so let's use a prime to indicate this distinction: $$ hat{bf e}'_i = R hat{bf e}_i. $$ From this it follows that $$ hat{bf e}_i = R^{-1} hat{bf e}'_i. $$ This fact can be useful to note, but for present purposes it is more useful merely to note that the old basis vectors $hat{bf e}_i$ can themselves be written in terms of the new basis vectors as a linear sum. So each $hat{bf e}_i$ is equal to some sum of $hat{bf e}'_i$ and we shall find $$ {bf u} = u'^1 hat{bf e}'_1 + u'^2 hat{bf e}'_2 +u'^3 hat{bf e}'_3 $$ where $u'^i$ are a new set of coefficients, called components. The point being that the vector $bf u$ has not changed but the components $u'^i$ are different from the components $u^i$ because the basis vectors $hat{bf e}'_i$ are different from the basis vectors $hat{bf e}_i$.

An example of this fact, now in 4 dimensions, is the widely used relation $$ u'^a = Lambda^a_{;mu} u^mu $$ where $Lambda^a_{; b}$ is the Lorentz transformation and I am now adopting the Einstein summation convention. In general relativity this gets generalised to $$ u'^a = frac{partial x'^a}{partial x^mu} u^mu. $$

So far we have maintained a strict distinction between a vector $bf u$ and its components $u^a$ or $u'^a$. But often it is useful to find a less cluttered notation, where indices (whether superscript or subscript) are not needed. So to this end let's define $$ U = { u^a }. $$ This equation asserts that the symbol $U$ refers to the set of components of the 4-vector $bf u$ with respect to the unprimed coordinate basis. Notice I am now using $U$ for the set of components and $bf u$ for the 4-vector. Next let's define $$ U' = { u'^a }. $$ This asserts that the symbol $U'$ refers to the set of components of the 4-vector $bf u$ with respect to the primed coordinate basis. Now if we are doing special relativity where the transformation is a Lorentz transformation, then we can conveniently write the relationship between $U$ and $U'$ as: $$ U' = Lambda U $$ where it is understood that the components of $Lambda$ are gathered together in a $4 times 4$ matrix, and the lists of components $U$ and $U'$ are to expressed as the components of column vectors, and $Lambda U$ is the ordinary matrix multiplication operation. This is quite a convenient notation so it is widely used, but it leads to the confusion between the set of components and the 4-vector itself. In a passive transformation, such as a change of inertial frame of reference, the 4-vector itself does not change, but the set of components changes from $U$ to $U'$. So strictly one should not call $U$ here 'the 4-vector' but rather 'the set of components of the 4-vector with respect to the unprimed inertial frame', and $U'$ is the set of components of the 4-vector with respect to the primed inertial frame.

If you don't like this index-free notation you don't have to use it. For special relativity I think it is quite a nice notation, but I would not employ it in G.R. where I find it clearer to stick to the use of indices when one is referring to components. This does not prevent one from referring directly to the 4-vectors or other tensors when one wishes.