The Book of Programming Ideas document contains 64 example programming exercises for students to try. The sections cover everything from basic input and output exercises through to subroutines, functions, and file handling. There are Python and Java versions of the document, although all exercises are written to be language-independent.
Subprograms are known by several different names - methods, functions, procedures, subroutines, and more. This video explains the different types of subprograms, including those that allow data to be passed in (as parameters) and those that return a value (functions). Even if you know how to create such code, reviewing the theory can be helpful.
CS Field Guide has an extremely comprehensive page about searching and sorting algorithms, including clear explanations, animated examples, and interactives. It covers algorithm efficiency and cost.
This videos provides an introduction into data mining, explaining why analysis of big data is necessary and how can we go about finding hidden patterns in large volumes of data. The end of the video includes some good examples of data analysis in the medical field, and the difficulties of finding correlation and causation with the data sets.
This is another interactive to help students learn about sorting algorithms. They must use a set of virtual scales to test and compare the weights of 10 jars. Students must then line the jars up in the correct order at the bottom of the page. There are, of course, multiple ways to solve the problem - some more efficient than others. CS Field Guide have a similar game.
The linear search or sequential search algorithm is very straightforward, but this video explains it. It also makes an important point that students often forget: that is, while we can see all of the numbers on display at the same time in our examples, a computer cannot. Therefore no algorithm can make 'jumps' or 'assumptions' about the data - it must be searched item by item if it is unsorted. The video also includes pseudo code for the algorithm
This Bubble Sort video does exactly as its name suggests - explaining the algorithm clearing with the aid of diagrams. Although it is labelled as "Java", it is relevant to all computer science students.
There is also a good explanation of the algorithm (including why it is very inefficient) plus step by step diagrams here.
Harvard's CS50 Merge sort video uses a really clear approach - numbered plastic cups which are moved around as we step through the algorithm. The same approach is used in all of their sorting algorithm videos, making it easy to compare different sorting algorithms.
Joe James' Merge sort algorithm video is also very clear.
For Merge sort in particular, step by step diagrams can be very helpful in understanding the algorithm. TutorialsPoint has a good set of diagrams, plus pseudocode.