How to use an instrumental variable to estimate the parameter? I have the following linear model of log wages (w) explained using years of schooling (S), years of experience and its square $(E,E^2)$ and 3 dummy variables indicating whether the individual was black (B), lived in the south (Sth), and lived in a metropolitan area (Sm).
$w_i=\alpha+\beta_1S_i+\beta_2E_i+\beta_3E_i^2+\beta_4B_i+\beta_5Sth_i+\beta_6Sm_i+\epsilon_i$ where $\epsilon_i$is an unobserved error term and $i$ indexes individuals.
A reduced form model for schooling is written as 
$S_i=\delta+z_i'\pi+v_i$ where $z_i$ is a $L\times 1$vector including the instrument (lived near college) and all the exogenous variables in the first regression equation, $\pi$ is a $L\times 1$vector of unknown parameters, $\delta$ is an unknown scalar parameter, and $v_i$ is an error term.
Assuming that $S_i$ is endogenous, how can I use the reduced form equation in conjunction with the first regression equation to identify the parameter $\beta_1$? How does the use of instrumental variable differ from the use of OLS estimator in this case?
Reduced form is a regression of dependent variable  on instrument directly without using some two stage approach.

Consider the following example of endogenous system 

Second Stage: $$Y = \alpha + \beta X + \epsilon$$

First Stage: $$X = \mu + \gamma Z+ \eta $$

Where $Y$ is dependent variable $X$ endogenous regressor and $Z$ is your instrument. 

One option is to estimate this with some two stage approach as previous post suggests where you would first estimate predicted $\hat{X}$ and then use the $\hat{X}$ in second stage instead of actual $X$ - however, this is not reduced form.

A reduced form is when you use the instrument directly in stead of the endogenous variable so in this case reduced form would be:

$$Y= \kappa + \omega Z + \xi$$

You can work from this general case to your specific one.

I would recommend looking at the handbook Mostly Harmless Econometrics form Angrist and Pischke. They discuss reduced form in Ch4 instrumental variables in action. 

The difference between OLS and IV estimator (in matrix notation) is as follows:

OLS: $\hat{\beta} = (\sum_i X_i X_i^{\prime} )^{-1}\sum_i X_i y_i$ - This estimator is biased in the presence of endogeneity

IV: $\hat{\beta}_{2SLS} = (x^{\prime} Z(Z^{\prime}Z)^{-1}x)^{-1}x'Z(Z^{\prime}Z)^{-1}y$ - (here $x$ is given by $x= Z \gamma + \eta$) this estimator has still bias but the bias declines if the first stage is strong - has high F-statistics as in approximation the bias term gets multiplied by $1/(F+1)$ where F is the test statistics so bias goes to zero as $F$ goes to infinity and in practice its very close to zero for $F&gt;10$.