Really, what Brett and Emblis have said here is more than sufficient. I'm not sure why I'm answering, but I guess it seemed like fun.
In my experience, relational databases (like SQL) have become essentially synonymous with data storage and databases. When somebody says "database", they're almost always referring to some sort of SQL database.
You will not be wasting time to learn SQL, and that w3schools link is an excellent starting point. I like Postgres and MySQL for learning. They're both simpler than say… Oracle. And they both can support you as far into the future as you need to go. There's no real need to leave MySQL behind as you get bigger and bigger. Postgres and MySQL can handle it all.
Brett's point is also good about MS SQL Server. It integrates better with C# through things like Entity Framework, which is an Object-Relational Mapping tool. (You give it objects, it figures out how to stuff them into the database.) The concept of an ORM sounds good on paper, and they're also worth learning about and understanding, but the reality is, they're not a silver bullet. At work, we've been significantly extending our logging database, and EF has proven more painful than helpful over the last few weeks. On a different day in a different month, I might feel differently about it, but today I'm just frustrated with it.
As you learn about SQL, don't forget to learn about good principles of database schema design. It's easy to push data into a DB and pull it out. But it takes more skill to get your DB structure correct and to sort out primary and foreign keys, etc. I'd say go find a good book on database design if you get serious about learning it. It will make a big difference. So as an example, look into things like database normalization.
I should also point out that there's been a bit of a backlash against SQL databases lately. There's a whole movement call the NoSQL Movement, and they have NoSQL databases. CouchDB and MongoDB are two systems that are popular in this category, though I'm sure there's plenty more. I think at least a part of this is that if you're using an object-oriented programming language like C#, the structure of SQL databases doesn't quite jive with objects, especially when it comes to things like inheritance. You can make it work, but it starts to get ugly fast. This is called the object-relational impedance mismatch. It's one of the reasons I've contemplated learning more about object databases (NDatabase caught my eye), though I haven't done enough with that to give you any meaningful feedback.
Having said that, I'd say that most databases around are still probably SQL databases, and probably will be for years or decades to come. They've been around this long, after all. At any rate, anything you learn in the SQL world will help you see programming from a different perspective, and none of it will be wasted. It's a great area to know about. Nearly every "real" job in software engineering will have some sort of persistent long-term storage attached to it. So it's a good skill to pick up.
