Model Driven Software Development (MDSD) is a style of software development that considers itself as an alternative to the traditional style of programming. The approach centers itself on building models of a software system. These models are typically made manifest through diagrammatic design notations - the UML is one option. The idea is that you use these diagrams, to specify your system to a modeling tool and then you generate code in a conventional programming language.
The MDSD vision evolved from the development of graphical design notations and CASE tools. Proponents of these techniques saw graphical design notations as a way to raise the abstraction level above programming languages - thus improving development productivity. While these techniques and tools never caught on too far, the basic core ideas still live on and there is an ongoing community of people still developing them.
Although I've been involved, to some extent, in MDSD for most of my career, I'm rather skeptical of its future. Most fans of MDSD base their enthusiasm on the basis that models are ipso facto a higher level abstraction than programming languages. I don't agree with that argument - sometimes graphical notations can be a better abstraction, but not always - it depends on the specific cases. Furthermore To use MDSD you need tools that support ProjectionalEditing, and these tools currently introduce a number of pragmatic issues in tooling - of which source control is the canonical example.
MDSD is surrounded by a terminological mess. One particular vision of MDSD is ModelDrivenArchitecture (MDA) which is an OMG initiative based on the UML. Many people in the MDSD community, however, don't think that MDA or UML is the right vision for MDSD. For a long time I would hear people talking about Model Driven Development (MDD) as the general concept and MDA as the OMG's specific vision. However the OMG has trademarks on several "Model Driven *" and "Model Based *" phrases - including MDD. As a consequence people have to be careful about what model driven phrase they use. I'm using MDSD as that is the title of a useful book on the topic.