This is the science and art of delivering the "best match" between a given user in a given context and a suitable advertisement (the right advertisement to the right user at the right time) in Online Advertising. It involves elements of learning and prediction.
For example Gmail uses computational advertising to decide which ads to serve to you based on the content of your current and all past emails.
A/B Testing is the process of finding the best ad or page element of two to deliver to the user and using statistical criteria to ensure the right choice.
Bandit Algorithms are a generalisation of A/B testing in two ways: (1) more than two ads can be tested and (2) the best ads are automatically delivered to users using a sequential learning process as the results come in. This is more efficient than the A/B method which chooses the best ad after the trial has ended.
The term "bandit" comes from the analogy of gambling on several slots machines ("one-armed bandits") where you simultaneously want to find the best paying machine and want to maximise profits as you do it.
This is a type of learning that learns as it goes without being presented in advance with previous data. An example is an algorithm that invests in the stock market without having the past history of stocks. The algorithm learns as it goes what the best stocks are and allocates more and more capital to them as it learns.
Another example is that of yacht racing where don't fully know the wind directions and strengths on the course but learn more and more each time you sail the course so each lap has a different optimum path even though wind conditions may not have changed.
Reinforcement Learning uses powerful algorithms such as Q-learning and TD-learning which are very specialised Machine Learning algorithms.
Much Computational Advertising involves analysing text and extracting information from it. An example is the Gmail application described above or a newspaper application which needs to put ads on a page of text. This is what creates the "context" part of the "given context" in the definition above.
A good algorithm should be able to figure out that an email with many mentions of the word "surfboard" should be matched with ad an for wetsuits even if the text didn't mention the word "wetsuit."
A Decision Tree is a set of decision rules for specifying Computational Advertising delivery. For example a tree might say: If the user has been here before then ... else if the user is using a Macintosh then ... etc.
The science is in calculating the optimum decision tree that maximises the response to the ad delivery (or ROI).
Personalisation refers to the customisation of the web site for individual users. It may be used for more than just ad delivery. For example, it may rank news items of interest to a news site reader.
Game Theory is the study of optimum strategic behaviour between two opponents. This and the similar science of Auction Theory comes into Computational Advertising when you have advertisers bidding against each other for ad space on your web site. It also comes up in yacht racing in the context of match racing where you are trying to beat an opponent instead of just trying to get around the course the fastest.