There aren't necessarily a set of algorithms that are specific to web development. More like algorithms common to all programming. I think the job postings that want you to have a knowledge of algorithms are just looking for someone with a solid background in Computer Science or the theoretical side of things.
You can think of your quest to obtain one of these jobs as two separate goals:
- Become knowledgeable about commonly used algorithms.
- Become proficient in one or two web programming languages.
For the first goal, I would recommend learning the basics of at least the following set of algorithms:
- Sorting algorithms
- Searching algorithms
- Simple recursive algorithms
- Backtracking algorithms
- Divide and conquer algorithms
- Dynamic programming algorithms
- Greedy algorithms
- Brute force algorithms
For the second goal, I would recommend picking two web programming languages, say PHP and ASP.NET, and finding some tutorials that help you learn the basics of these. Then, try to build something useful using these and before you know it, you'll be ready to make a contribution to a software development team!