EPSG:4326 is a geographic coordinate system (GCS) based on latitude/longitude, a particular ellipsoid approximating the shape of the Earth, and a known location for the coordinate system origin (0 degrees longitude, 0 degrees latitude).
Mercator and Lambert Conformal Conic projections are methods for calculating map coordinates to be plotted on a flat surface from the latitude/longitude coordinates on a globe. Mercator and Lambert Conformal Conic are different methods for doing these calculations and when used they will project globe data in different ways resulting in maps that look different and have different uses. For a visual comparison see Mercator and Lambert.
Each projection results in it's own Cartesian coordinate space on the map, although maps will often still include latitude and longitude lines so map users can estimate GCS coordinates even though the map is actually drawn on a different coordinate system. This does get confusing!
If you need to use a Lambert Conformal Conic projection, you will select that specifically in whichever GIS you are using and possibly tweak parameters for the projection such as where you want the standard parallels. There are many variations of Lambert Conformal Conic projections, each optimized for different regions of the globe. You should be able to search for "Lambert" (or "lcc" if using software that relies on Proj) in most GIS projection interfaces to get a fairly short list of those available. For example, here is one list of Lambert conic projection variants for Canada.