A clear introduction to what both of these are:
1. Tableau Desktop:
It is a self service business analytics and data visualization that anyone can use. It translates pictures of data into optimized queries. With tableau desktop, you can directly connect to data from your data warehouse for live upto date data analysis. You can also perform queries without writing a single line of code. Import all your data into Tableau’s data engine from multiple sources & integrate altogether by combining multiple views in a interactive dashboard.
2. Tableau Server:
It is more of a enterprise level Tableau software. You can publish dashboards with Tableau Desktop and share them throughout the organization with web-based Tableau server. It leverages fast databases through live connections.
Basically you can get both of them or just get the license of Tableau Desktop and the viewers of the dashboard can use Tableau Reader which is a free software.