Logit and probit
Logit and probit models are special cases of regression designed to deal with binary outcome variables. Some examples of binary outcome variables (aka dichotomous outcome variables), often coded as 0/1: a candidate won or lost an election, a plant was or was not observed in an experimental plot, a rat did or did not take a left turn in a maze.
The difference between probit and logit models lies in the underlying model for the regression. In the logit model (logistical regression), "the log odds of the outcome is modeled as a linear combination of the predictor variables."  In the probit model, "the inverse standard normal distribution of the probability is modeled as a linear combination of the predictors." 
note: Like the rest of the D@R help pages, the goal of this documentation is not to help you choose what test to run or to teach you statistics, but to help you through the mechanics of running statistical tests. For help deciding what sort of analysis to run on your data, consult with your advisor/professor. (This chart may prove useful as you are considering your options.)
The links below contain worked examples, annotated output, and documentation from Stata. If you run into issues with your code, email the Data@Reed team -- we're here to help.
Stata documentation: logit
Stata documentation: probit
References / sources
 UCLA/IDRE page: logistical regression
 UCLA/IDRE page: probit regression