ito vs Stratonovich

The Stratonovich integral obeys the usual chain rule when performing change of variables, so can be easier to use to perform some calculations. The Itō integral, on the other hand, is a martingale, which lends it some nice theoretical properties—and nice theorems for taking advantage of them. See this related answer for more info.

Usually adapted processes are said to not "see into the future". Both the Itō and Stratonovich integrals are adapted processes and thus do not see into the future.

Both are L2 and probability limits of Riemann sums (as the size of the subdivision goes to 0).

Ito integral

$$\sum X_{t_{i-1}} (S_{t_i} - S_{t_{i-1}})$$

Imagine one controls the $X$ process who does not know the future of the process $S$. This is typically the case for mathematical finance, where $S$ is the value of an asset and $X$ is the number of shares, the integral being the value of the investment in $S$.

Stratonovich integral

$$\sum\frac12 (X_{t_i} + X_{t_{i-1}}) (S_{t_i} - S_{t_{i-1}})$$

The advantage of this integral is that if $f$ is smooth enough, you keep the standard chain rule for derivation for $f(X_t)$:

$$df(X_t) = f'(X_t) dX_t$$

It is not the case with the Ito integral: there is a second order term to correct the previous formula.