Eisenhower was most clearly a 5-star General of the Army distinct from a General of the Armies of the US - you need a specific Act of Congress to be a GOTASUS. Among the Western allies, the equivalent-ranking people, i.e. clearly 5 steps above Colonel, were the Field Marshals in the Commonwealth/Empire forces. Marhsall, Ike, etc. were made to have a rank equivalent to Montgomery and fellows’s and Ike in turn was* appointed to be SACEUR. He had authority over all others by virtue of his post*, just as in other circumstances you have 4-star generals and admirals being supreme commanders over *other *4-star generals and admirals.
Field Marshal is a historic grade that evolved to survive to this day. As the words indicate, it’s the officer that marshals the forces in the field (thus the colloquial sports usage to refer to a gridiron quarterback). As the general grades became more complexly defined, it rose in status as most Old-World nations reserved the title for the seniormost generals. Some, e.g. German armies up to WW2, as a true operational rank – Generalfeldmarschal as the superior to Generaloberst (colonel general). Others as a special dignity that is granted to a general for very distinguished service or when holding a special post (France).
“Brigadier” referred to the rank level at which you’d be in charge of a Brigade (duh), a unit assembled of components from more than one regiment, not yet a division; currently it’s the main maneuver component of a division - usually at least two batalllions of your primary combat arm plus appropriate units of other branches (say, a cavalry troop, an engineer company, etc.).
Depending on your military’s history, the traditional officer grade associated with this command was either a senior colonel or other super-field-grade officer (as in the British “Brigadier”, known to Dr. Who fans) or the lowest rank of General. Napoleonic armies organized generalships by command: Brigade General, Division General, Corps General, Army General; many nations borrowed part of this structure - English speaking nations generally only the bottom grade because they tended to not have had it before (to this day, some armies do NOT have a BG equivalent and skip directly from COL to MG); Spanish-speaking armies have tended to borrow the bottom two. In modern practice, US Brigades are actually commanded mostly by full colonels.
The straight jump from COL to MG in some armies created more room at the top, actually – in the Soviet military the grades were COL, then MG, LTG, Colonel General, Army General, and then **two **different types of Marshal: “Of (Branch)” and “Of The Soviet Union”.