LookDev stands for "Look Development", it is the equivalent of the Material preview mode in 2.7# with better shading and more advanced features.
This is a viewport mode designed to develop the look and appearance of your materials and shaders, it has realistic lighting and material previews are represented textured, fully shaded, and as close to the final rendered result as possible.
It differs from Rendered mode in that it doesn't necessarily use your scenes native lighting and environment, instead it (optionally) overrides scene lamps and scene world with a generic user configurable environment lighting, that is both more neutral on your materials and quicker to render, yielding faster performance for instant previews and more responsive editing.
It is harder to understand its value when working with EEVEE. Since differences to Render Mode are tenuous and blurry, and given how both are based on the same OpenGL real time rendering engine, it may seem like a pointless and redundant feature.
Its true purpose becomes more apparent when rendering with Cycles, where the Rendered viewport mode is actually rendered by Cycles based path tracing. While impressively quick it is too slow for preview and unsuitable as a working mode while editing; while LookDev is using EEVEE based preview in background trying to match Cycles output as close as possible with responsive OpenGL based rasterization.
For the time being, to render the current viewport result at any time go to the 3D View header under View > Viewport Render or View > Viewport Render Animation
I'm sure eventually there will be dedicated render buttons in the render panel to do this, both for EEVEE and OpenGL render.