Model Resolution and Spherical Harmonics
Gridded data on a sphere can instead be represented by a series of spherical harmonic functions. This can both reduce data storage and computation as well as make some calculations easier to perform. The series is generally truncated to some zonal and meridional wavenumbers. The higher the numbers, the higher the spatial resoltuion retained. Two common truncations are triangular where the zonal and meridional wave numbers retained are the same and rhomboidal where the zonal and meridional add up to a constant. More information can be found in NCAR's description of the ERA-40 dataset, and a nice illustration from euromet.
The ECMWF spectral model is based on a triangular truncation (indicated by TLM where M is the maximum zonal wavenumber). The model employs reduced Gaussian grids for all computations in physical space. A Gaussian grid has a variable spacing of latitude circles which is optimized for transforms to and from physical space that are needed in the model calculations. In a reduced Gaussian grid the number of grid points per latitude circle is reduced as the distance from the equator increases, which results in a more uniform distribution of grid points on the sphere.
More details can be found in the ECMWF training course notes. For Gaussian grids, the resolution is indicated by NX where X is the number of latitude circles between equator and pole. Information about grid point spacing for various Gaussian grid resolutions used at ECMWF can be found here.