What is the difference between two stage least squares and instrumental variable regression? I'm doing independent study and I am having trouble understanding the difference between these two estimators.

I get that 2SLS is predicting the endogenous variable, and that instrumental variables are similar to proxy variables, but I don't get how one differs from the other.
The meaning of the words first

Some people use the word "IV estimator" to refer to any estimator that uses instrumental variables. To them, IV estimators contain 2SLS, LIML, k-class estimators, and others, so 2SLS is a special case of IV. For example, the title of Bekker's (1994, Econometrica) paper is "Alternative approximations to the distribution of instrumental variable estimators".

More traditional people mean by IV the particular instrumental variable estimator $(Z'X)^{-1}Z'y$ for the exactly identified case ($Z$ = instrument matrix, $X$ = regressor matrix, $y$ = regressand vector), and 2SLS is a generalization of IV to the overidentified case. But, as Paul says, 2SLS can be expressed as an IV estimator of this second sense because it is $(\hat{X}'X)^{-1} \hat{X}'y$, where $\hat{X} = Z(Z'Z)^{-1}Z'X$ is the instrument matrix.

I personally think it is very fine to leave the meaning of IV estimators ambiguous because the meaning is usually clear in the context and we need not rigorously distinguish them.

As for your question ...

It seems to me that the sentence "2sls is predicting the endogenous variable" means the first stage regression of the endogenous regressor on the instrumental variables (to get $\hat{X}$). The expression "instrumental variables are similar to proxy variables" looks more casual. Proxy variables (e.g., IQ for ability) can be used to solve the endogeneity problem. Instrumental variables are another way of solving the endogeneity problem. In that sense they are "similar".