Hilbert spaces, briefly
A formal mathematical setting for a quantum system is that of a Hilbert space \mathcal{H}, which is (for us) just a vector space along with an inner product.
Given a Hilbert space corresponding to our system, the result of any preparation of the system is then represented by some unit vector |\psi\rangle\in \mathcal{H}, and any test is represented by some other unit vector |e\rangle\in \mathcal{H}.
The inner product of these two vectors, \langle e|\psi\rangle, gives the probability amplitude that an object prepared in state |\psi\rangle will pass a test for being in state |e\rangle.
As always, probabilities are obtained by squaring absolute values of probability amplitudes:
|\langle e|\psi\rangle|^2
= \langle\psi|e\rangle\langle e|\psi\rangle.
After the test, in which the object was found to be in state |e\rangle, say, the object forgets about its previous state |\psi\rangle and is, indeed, actually now in state |e\rangle.
That is, if we immediately measure the object again, we will find it to still be in state |e\rangle with probability 1.
This is the mysterious quantum collapse, which we will further discuss later on.
A more complete test involves multiple states e_k that form an orthonormal basis \{|e_1\rangle,\ldots,|e_n\rangle\}.
These states are perfectly distinguishable from each other: the condition \langle e_k|e_l\rangle = \delta_{kl} implies that a quantum system prepared in state |e_l\rangle will never be found in state |e_k\rangle (unless k=l).
The probability amplitude that the system in state |\psi\rangle will be found in state |e_k\rangle is \langle e_k|\psi\rangle and, given that the vectors |e_k\rangle span the whole vector space, the system will be always found in one of the basis states, whence
\sum_k |\langle e_k|\psi\rangle|^2 = 1.
As a result:
A complete measurement in quantum theory is determined by the choice of an orthonormal basis \{|e_i\rangle\} in \mathcal{H}, and every such basis (in principle) represents a possible complete measurement.