the complexity-class map
Step back from the individual definitions and a landscape comes into view: a nested chain of classes, each contained in the next, mapping out how time and memory budgets relate. The headline chain, from least to most resource, is L contained in NL contained in P contained in NP contained in PSPACE contained in EXPTIME. Reading it is like reading a series of ever-larger circles drawn one inside another, each ring a more generous resource budget than the one within it. This single picture organizes almost everything in the field.
Each inclusion has a one-line reason. L is contained in NL because determinism is a special case of nondeterminism. NL is contained in P because a log-space machine has only polynomially many configurations and reachability among them is solvable in polynomial time. P is contained in NP because a verifier can ignore its certificate and just recompute. NP is contained in PSPACE because you can try every candidate certificate one at a time, reusing the same memory. PSPACE is contained in EXPTIME because polynomial space allows only 2^(poly) configurations, so a non-looping machine halts in exponential time. Layered around NP sit co-NP and the polynomial hierarchy (the Sigma and Pi levels), all inside PSPACE; and parallel space classes extend the picture (PSPACE inside EXPSPACE, and so on).
Now the brutal honesty, which is the whole point of this map. We know the two ENDPOINTS differ: by the hierarchy theorems, L is strictly inside PSPACE and P is strictly inside EXPTIME, so the chain genuinely grows somewhere. We know NL = co-NL (Immerman-Szelepcsenyi) and PSPACE = NPSPACE (Savitch), pleasant collapses. But almost every individual inclusion in the middle is CONJECTURED, not proved: whether L = NL, whether L = P, whether P = NP, whether NP = PSPACE, are all famous OPEN problems. It is entirely possible (though wildly disbelieved) that L all the way up to PSPACE are secretly one and the same class. The map shows what must be true and what we wish were true; distinguishing the proven from the merely believed is the deepest unfinished business of the subject.
The known-strict facts you can lean on: by the space hierarchy theorem L is strictly inside PSPACE, and by the time hierarchy theorem P is strictly inside EXPTIME. So somewhere along L inside NL inside P inside NP inside PSPACE at least one inclusion must be strict, and likewise within P inside NP inside PSPACE inside EXPTIME, but we cannot yet point to WHICH one.
L in NL in P in NP in PSPACE in EXPTIME: the endpoints provably differ, but almost every middle inclusion is open.
Most inclusions in the chain (L vs NL, L vs P, P vs NP, NP vs PSPACE) are CONJECTURED, not proved. Only the endpoint separations (hierarchy theorems: L strictly inside PSPACE, P strictly inside EXPTIME) and the collapses NL = co-NL, PSPACE = NPSPACE are actually known.