Difference between "information" and "useful information" in algorithmic information theory
The central concept here is Kolmogorov complexity, and more specifically compressibility. To get a intuitive feeling of compressibility, consider two strings $A \in \mathbb{B}^*$ and $B \in \mathbb{B}^*$, where $\mathbb{B} = \{ 0,1 \}$. Let

$A = 1010$ $1010$ $1010$ $1010$, and
$B = 1011$ $0110$ $0111$ $1001$.

Note that $|A| = |B| = 16$. How could we quantify how much information $A$ or $B$ has? If we think about classical information theory, in general, transmitting a string of length $n$ takes $n$ bits on average. However we cannot say how many bits we need to transmit a specific string of length $n$.
Why is the information content of a random string not zero?
On a closer look, we can see that in fact $A = 10^8$. However, it is much harder to say if $B$ has any obvious patterns in its structure, at least it seems and feels more random than $A$. Because we can find a pattern in $A$, we can easily compress $A$ and represent it with less than $16$ bits. Likewise, since it is not easy to detect any patterns in $B$, we cannot compress it as much. Therefore we can say that $B$ has more information than $A$. Moreover, a random string of length $n$ has maximal information since there is no way we can compress it, and hence represent it with less than $n$ bits.
What is useful information, then?
For useful information, yes, there is a definition using a Turing machine $T$. The useful information in $x \in \mathbb{B}^*$ is
$$\min_T \space \{\space l(T) + C(x|T) : T \in \{ T_0, T_1, ... \} \},$$
where $l(T)$ denotes the length of a self-limiting encoding for a Turing machine $T$. The notation is usually such that $C(x)$ denotes the Kolmogorov complexity of $x$ and $C(x|y)$ the conditional Kolmogorov complexity of $x$ given $y$.
Here $T$ embodies the amount of useful information contained in $x$. What we could ask is which such $T$ to select among those that satisfy the requirement. The problem is to separate a shortest program $x^*$ into parts $x^* = pq$ s.t. $p$ represents an appropriate $T$. This is actually the very idea that spawned minimum description length (MDL).