Procrastinator has given you a story about Higgs Boson, but that really doesn't explain how decision trees and boosting enters into it. I will give you a general description of these classification algorithms. The classification problem involves taking groups that are categorized and a set of data with several variables (called features) that describe the points. The idea is that points from different categories will have very different combinations of values for these features. Given a set of points with known categories and associated features the classification algoithm will assign a catgory based on the value of the features. This is done to identify the category for a new point whose category we do not know.
The original data with the categories known is called training data and is used to construct the classification rule. A decision (or classification tree) is a binary structure shaped like the branches of a tree. There are two branches at each node. The baranches are constucted so that given certain values of the feature you take one branch and for the remaining values you take the other each branch leads to a node where additional branches are formed.
At each node branches are form using one of the features. This is followed down to terminal nodes at which the category is selected. The order of the the features on the tree and the way the values are split down the branches are determined based on the training data in order to correctly classify the training data accurately. Accuracy is determined by applying the rule given by the tree to test data. There are potentially many trees that can do a good job at classifying the points.
In boosting several trees (called an ensemble) are constructed and the object is classified by all the trees. Then a rule such as majority vote is used to make the decision. This is called ensemble averaging and the procedure is called boosting because it has been shown that generally this procedure produces a more accurate classification rule than any individual tree would do on its own.