Algorithmic complexity is a formal measure of the minimum amount of information required to specify some message, binary string, computer program, classification rule, etcetera. Roughly, the algorithmic complexity or Kolmogorov complexity of a message is the number of 1s and 0s required to specify a Turing machine that reproduces the message. Similarly, the algorithmic complexity of a classification rule is the length of the binary string required to encode a Turing machine that faithfully carries out the classification rule.
(The relevant Wikipedia article is filed under Kolmogorov complexity. Not to be confused with "computational complexity", which isn't the same thing at all.)
A string of a million 1s has around as much K-complexity as the number 1,000,000, since getting a Turing machine to print out exactly a million 1s and then stop, is mostly a matter of encoding the number 1,000,000 (which itself can potentially be compressed further, since it's an unusually simple number of that magnitude: namely, it's 10 to the 6th power, so if we already have a concept of 'exponentiation' or can encode it simply, we just need to encode the numbers 10 and 6, which are quite small).
When we say that a message has high Kolmogorov complexity, we mean that it can't be compressed beyond a certain point (unless the 'compression algorithm' is itself very large and secretly contains a lot of the key information baked in). Things have high Kolmogorov complexity when they're made up of many independent facts that can't be predicted from knowing the previous facts.
Shakespeare's Romeo and Juliet will compress quite a lot using simple algorithms, because there are many words used more than once, and some words are much more frequent than others. But we are exceedingly unlikely to find, anywhere in the first trillion Turing machines, any Turing machine that prints out the exact text of this play. 2^40 is greater than a trillion, so if we consider the set of all 40-bit binary strings, it's clear we can't print out all of them exactly using the first trillion Turing machines. Finding a 40-bit Turing machine that printed out the exact text of Romeo and Juliet would be vastly improbable. We can also observe that a program that prints out all possible books is much simpler than a program that prints out only Romeo and Juliet.
On the other hand, it wouldn't be particularly surprising to find a small Turing machine that printed out 1s and then stopped, because the algorithm for this enormous number seems simple, and easy to encode as a computer program or Turing machine.