LookDev stands for "Look Development", it is the equivalent of the Material Preview mode in 2.7# with better shading and more advanced features. In fact as of Blender 2.82 it has actually been renamed back to Material Preview.
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 as possible to the final rendered result with real time emulation.
It differs from Rendered mode by not necessarily using 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 produced by Cycles based path tracing. While impressively quick, it may still be too slow for preview and is unsuitable as an actual working mode while editing; while LookDev is using EEVEE based preview that is trying to match Cycles output as close as possible with responsive OpenGL based rasterization.
To render the current viewport result at any time go to the 3D View header menu under View > Viewport Render or View > Viewport Render Animation
To get pre-2.8 like render buttons in the Properties Window like in 2.79 you can install one of several available addons like Render Buttons to bring back the old workflow.