The Batch API is a powerful feature in Drupal that allows complex or time consuming tasks to be split into smaller parts.
For example, let's say you wanted to run a function that would go through every page on you Drupal site and perform an action. This might be removing specific authors from pages, or removing links in text, or deleting certain taxonomy terms. You might create a small loop that just loads all pages and performs the action on those pages.
This is normally fine on sites that have a small number of pages (i.e. less than 100). But what happens when the site has 10,000 pages, or a million? Your little loop will soon hit the limits of PHP execution times or memory limits and cause the script to be terminated. How do you know how far your loop progressed through the data? What happens if you tried to restart the loop?
The Batch API in Drupal solves these problems by splitting this task into parts so that rather than run a single process to change all the pages at the same time. When the batch runs a series of smaller tasks (eg. just 50 pages at a time) are progressed until the task is complete. This means that you don't hit the memory or timeout limits of PHP and the task finishes successfully and in a predictable way. Rather than run the operation in a single page request the Batch API allows the operation to be run through lots of little page request, each of which nibbles away at the task until it is complete.
This technique can be used any a variety of different situations. Many contributed modules in Drupal make use of this feature to prevent processes taking too long.