Data mining is one part of the process of Knowledge Discovery in Databases.
There are many techniques within data mining that aim to accomplish different tasks. Generally tasks fall into one of two categories, predictive or descriptive. Predictive tasks look at historical data to predict what will happen in the future. Descriptive tasks will look at some given data and find patterns in it.
Since data mining is a growing area, the techniques are constantly changing, as new improved methods are discovered. At present, some of the most well known predictive algorithms, known as classification algorithms include Naive Bayes, SVM, Decision Trees (such as C4.5), Artificial Neural Networks, k-Nearest Neighbour and more. Some predictive algorithms are able to perform regression, a form of prediction for non-categorical data. Some of the most well known descriptive algorithms include the Apriori and FP-tree algorithms (for finding association rules), K-Means and Hierarchical clustering algorithms, GSP and PrefixSpan for Sequential Pattern Mining and various algorithms for Outlier Detection. In 2006, at the International Conference on Data Mining (ICDM), the top algorithms were discussed (see http://www.cs.uvm.edu/~icdm/algorithms/index.shtml).
This is a very limited list and many more algorithms have been and are being developed, as this area continues to grow and expand to encompass new problems and applications.