How do you know when two objects are entangled? I’m not asking how would you entangle two objects. I want to know how would you know they are entangled?
 A: *

*If you only have a single copy of the system, then the task is impossible - you simply cannot tell.

*If you have a large ensemble of copies of the system, but you don't know what state they're in, then you need to find out what that state is. That task is known as quantum state tomography, and it comes in a variety of flavours depending on what the system itself is. Generically, this will involve taking a number of correlated measurements in several incompatible bases on both sides of your bipartite state, plus some additional post-processing of those measurement results to reconstruct the joint quantum state.
Once you do have that quantum state, there are a bunch of different entanglement measures. If your state is pure, then things are relatively simple, and all the different measures will coincide, but for mixed states things are fairly complicated and there is a nontrivial hierarchy of non-equivalent measures of entanglement. Some keywords to look for: entropy of entanglement, entanglement of formation, distillable entanglement, etc.

*If you do know what state you're meant to have, then things are a bit simpler, and you can use an entanglement witness. An entanglement witness is an observable $\hat W$ such that its expectation value
$$\langle W \rangle = \mathrm{Tr}(\hat W\hat \rho)$$
can only be negative if the state $\rho$ is entangled (i.e. it is not separable, i.e. if its correlations are strictly beyond the classical "left-shoe-right-shoe" correlations).
Unfortunately, there are no universal entanglement witnesses: if $\hat \rho$ is entangled then it is guaranteed to have at least one entanglement witness $\hat W$ such that $\mathrm{Tr}(\hat W\hat \rho)<0$, but not all entanglement witnesses will work for all entangled states. (Hence the need for knowing what the state is for producing the right witness.)
A: In general you can't. That is, if you have just two particles you cannot tell whether they are entangled or not.
Entanglement reveals itself by correlations. For example if you take many pairs of particles you may find that their properties are always correlated, e.g. their spins are always opposite, and this tells you that whatever mechanism is generating the pairs of particles is entangling them. But this shows up only with repeated measurements. A single measurement cannot tell you the particles are correlated since their spins could have the values you observe just by chance.
A: One way to know two objects are entangled is to prepare them that way, say by using an SPDC apparatus.  See Andrew Steane's answer for a better way.
A: Partial answer:
If you can measure entropy of a collection of identically prepared systems, then you could tell if the sub-systems are correlated.
Suppose the whole system is in a pure state $|\psi_{\text{tot}} \rangle$.  The whole system density operator is defined as
\begin{equation}\tag{1}
\rho_{\text{tot}} = |\psi_{\text{tot}} \rangle \otimes \langle \psi_{\text{tot}}|.
\end{equation}
The state of the two sub-parts is defined by a partial-trace:
\begin{align}
\rho_1 &= \mathrm{Tr}_2 \, \rho_{\text{tot}}, \tag{2} \\[12pt]
\rho_2 &= \mathrm{Tr}_1 \, \rho_{\text{tot}}. \tag{3}
\end{align}
Then compute their respective entropy:
\begin{align}
S_1 &= -\, \mathrm{Tr}(\rho_1 \, \ln{\rho_1}), \tag{4} \\[12pt]
S_2 &= -\, \mathrm{Tr}(\rho_2 \, \ln{\rho_2}). \tag{5}
\end{align}
Entropy of the whole system is 0 since it's a pure state:
\begin{equation}\tag{6}
S_{\text{tot}} = -\, \mathrm{Tr}(\rho_{\text{tot}} \, \ln{\rho_{\text{tot}}}) = 0.
\end{equation}
In general (even when the whole system is in a mixed state):
\begin{equation}\tag{7}
S_{\text{tot}} \le S_1 + S_2.
\end{equation}
If there's no correlations between both sub-systems, then $S_{\text{tot}} = S_1 + S_2$ ($S_1 = S_2 = 0$ if the sub-systems are in a pure state).
You can say that both sub-systems are correlated when $S_{\text{tot}} < S_1 + S_2 \ne 0$.
However, I'm not sure that entangled is synonymous to correlated.
A: There are already good answers here; I'm going to expand on one aspect.
Two answers point out, correctly, that with only one pair of particles and no prior knowledge, you cannot discover whether or not they were entangled before you did anything with them such as measure them.
Then a couple of answers show what you can do if you have access to many copies which you have reason to know are all in the same state. You can use entanglement witnesses and/or tomography.
One answer added a hint of something I will expand on briefly. This is that if you know the procedure whereby the pair of particles has been prepared, then you can know they are entangled. WillO mentioned the use of parametric downconversion. However that normally relies on a spontaneous process, with the result that the entangled pairs can only be inferred after the fact by post-selection, so I think it is not the best example to pick. A better example would be something like an experiment with a pair of atoms where you first prepare them in their ground state (or another suitable state) by optical pumping, and then apply to them an entangling sequence of operations such as a couple of suitably chosen laser pulses (something we have done about a zillion times in my lab, and it works fine!) 
So the logic of this latter method works is as follows. 


*

*Perform a fixed sequence of operations.

*Observe the outcome.

*Repeat, where in step 2 you carry out different observations each time, so as to build up information on what was the outcome of 1. e.g. After 1 million repetitions you have a lot of information (tomography) about what was the result of step 1.

*Now repeat the whole sequence (1,2,3), say about one thousand times. The idea here is to check that the set of almost-complete tomographic information acquired each time says the same (to within some precision).

*You are now ready to declare that, next time you press the button to perform step 1 (just that step, using just one pair of atoms), then the outcome will be that they are in an entangled state.
Finally, I would like to underline that in order to show the presence of entanglement it is not enough merely to show correlation in one direction. This is a basic, simple, beginner-level point, but it seems that there are a lot of people who want to express opinions about entanglement whose knowledge of the subject has not even grasped this basic point.
