Polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modelled as an nth degree polynomial in x. It's been used to describe nonlinear phenomena such as the progression of disease epidemics.
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Linear Regression is a linear approach for modelling the relationship between a scalar dependent variable y and one or more independent variables denoted X. The case of one independent variable is called Simple Linear Regression. For more than one independent variable, the process is called Multiple Linear Regression.
Random Forests are an ensemble learning method for classification, regression and other tasks, that operate by constructing a multitude of decision trees at training time and outputting the class that is the mode of the classes (classification) or mean prediction (regression) of the individual trees.
Random Forest is composed by many Decision Trees (default 10). Each tree in the forest considers a random subset of features when forming questions and only has access to a random set of the training data points. The Random Forest takes an average of all the individual Decision Tree estimates to make a prediction.