Inversion of Control is primarily about dependency management and providing testable code. From a classic approach, if a class has a dependency, the natural tendency is to give the class that has the dependency direct control over managing its dependencies. This usually means the class that has the dependency will 'new' up its dependencies within a constructor or on demand in its methods.
Inversion of Control is just that...it inverts what creates dependencies, externalizing that process and injecting them into the class that has the dependency. Usually, the entity that creates the dependencies is what we call an IoC container, which is responsible for not only creating and injecting dependencies, but also managing their lifetimes, determining their lifestyle (more on this in a sec), and also offering a variety of other capabilities. (This is based on Castle MicroKernel/Windsor, which is my IoC container of choice...its solidly written, very functional, and extensible. Other IoC containers exist that are simpler if you have simpler needs, like Ninject, Microsoft Unity, and Spring.NET.)
Consider that you have an internal application that can be used either in a local context or a remote context. Depending on some detectable factors, your application may need to load up "local" implementations of your services, and in other cases it may need to load up "remote" implementations of your services. If you follow the classic approach, and create your dependencies directly within the class that has those dependencies, then that class will be forced to break two very important rules about software development: Separation of Concerns and Single Responsibility. You cross boundaries of concern because your class is now concerned about both its intrinsic purpose, as well as the concern of determining which dependencies it should create and how. The class is also now responsible for many things, rather than a single thing, and has many reasons to change: its intrinsic purpose changes, the creation process for its dependencies changes, the way it finds remote dependencies changes, what dependencies its dependencies may need, etc.
By inverting your dependency management, you can improve your system architecture and maintain SoC and SR (or, possibly, achieve it when you were previously unable to due to dependencies.) Since an external entity, the IoC container, now controls how your dependencies are created and injected, you can also gain additional capabilities. The container can manage the life cycles of your dependencies, creating and destroying them in more flexible ways that can improve efficiency. You also gain the ability to manage the life styles of your objects. If you have a type of dependency that is created, used, and returned on a very frequent basis, but which have little or no state (say, factories), you can give them a pooled lifestyle, which will tell the container to automatically create an object pool for that particular dependency type. Many lifestyles exist, and a container like Castle Windsor will usually give you the ability to create your own.
The better IoC containers, like Castle Windsor, also provide a lot of extendability. By default, Windsor allows you to create instances of local types. Its possible to create Facilities that extend Windsor's type creation capabilities to dynamically create web service proxies and WCF service hosts on the fly, at runtime, eliminating the need to create them manually or statically with tools like svcutil (this is something I did myself just recently.) Many facilities exist to bring IoC support existing frameworks, like NHibernate, ActiveRecord, etc.
Finally, IoC enforces a style of coding that ensures unit testable code. One of the key factors in making code unit testable is externalizing dependency management. Without the ability to provide alternative (mocked, stubbed, etc.) dependencies, testing a single "unit" of code in isolation is a very difficult task, leaving integration testing the only alternative style of automated testing. Since IoC requires that your classes accept dependencies via injection (by constructor, property, or method), each class is usually, if not always, reduced to a single responsibility of properly separated concern, and fully mockable dependencies.
IoC = better architecture, greater cohesion, improved separation of concerns, classes that are easier to reduce to a single responsibility, easily configurable and interchangeable dependencies (often without requiring a recompilation of your code), flexible dependency life styles and life time management, and unit testable code. IoC is kind of a lifestyle...a philosophy, an approach to solving common problems and meeting critical best practices like SoC and SR.
Even (or rather, particularly) with hundreds of different implementations of a single interface, IoC has a lot to offer. It might take a while to get your head fully wrapped around it, but once you fully understand what IoC is and what it can do for you, you'll never want to do things any other way (except perhaps embedded systems development...)