Puuhhh, this is a very big question and there is no simple answer. But I will do my best to explain my own opinion at least:
What are containers?
Containers provide functionality to you. Such a functionality can be to handle web request and dispatch them to servlets - in this case we call them servlet containers (e.g. Tomcat or Jetty).
But containers can also provide other things, e.g. they can provide user authentication, logging or the connection to a database. Most containers (e.g. Tomcat) do multiple of those things (e.g. Tomcat does all I mentioned). Some containers do more then others, e.g. JBoss can do much more than Tomcat.
Trade Off
However, there is a trade off: If you use a simple container (like Tomcat), you need to do a lot of things on you own or by using other Frameworks (like Spring). But if you use a powerful container, you must know the container very well and the chance is high that your application will depend on this concrete container sooner or later.
The point, that using a container is not mandatory. It is a decision. Some people will argue for it, others against it. But depending on the books you read, this decision is already made (e.g. J2EE needs a J2EE container, that's how it works).
The trend (IMHO)
Years ago the trend was to use big and powerful (J2EE) containers which provide as much as possible. IMHO the trend today is to use smaller and light-way solutions. Most developers would prefer to use a Tomcat server instead of a JBoss server today.
Frameworks without containers
While J2EE needs a container, there are other frameworks/technologies which supports the development of web applications without any external container. Such frameworks are Play! or Spark Java.
Note
If you are not familiar with containers and Spring, take care to don't get confused. Most applications you will develop with Spring are web applications which will be deployed to a servlet container. This is very common. But Spring doesn't relay on that. You can also use Spring without such a container, e.g. to develop a desktop application. But if you want to develop a web application, the Java-way is to use a servlet container.