the Sobolev norm
If a Sobolev space is the room where rough functions live, the Sobolev norm is the ruler that measures how big each one is. But unlike an ordinary 'size', it weighs not just the function itself but also its slopes and curvatures — its derivatives up to the chosen order. A function with mild values but violently wiggling derivatives is judged large; the norm sees the wiggles, not just the height.
Concretely, the W^{k,p} norm of u adds up the L^p sizes of u and all its weak derivatives up to order k. For the central case p = 2 (the space H^k), the squared norm is a clean sum of integrals: ||u||_{H^k}^2 = integral of (u^2 + |grad u|^2 + ... + |k-th derivatives|^2) over U. For H^1 in particular, ||u||_{H^1}^2 = integral of u^2 dx + integral of |grad u|^2 dx — the second piece, integral of |grad u|^2, is the famous Dirichlet energy, the cost of bending. Because this H^k norm comes from an inner product (u, v)_{H^k} = integral of (u v + grad u . grad v + ...), the space is a Hilbert space, and that inner-product structure is exactly what powers Lax-Milgram and Galerkin. One also speaks of the seminorm, which keeps only the top-order derivative term and ignores u itself; on the right space (like H^1_0) the seminorm is already a full norm, by the Poincare inequality.
Why this matters: convergence in a Sobolev norm is much stronger than convergence in L^2. If u_n converges to u in H^1, then not only do the functions get close, their gradients get close too — so you cannot smuggle in ever-wilder oscillations while pretending to converge. This control of derivatives is precisely why a priori estimates in PDE are stated as bounds on Sobolev norms: bounding ||u||_{H^k} pins down both the solution and its smoothness, and the Sobolev embedding theorem then cashes a finite Sobolev norm out into honest continuity or boundedness.
Compare two functions on (0, pi): u(x) = sin(x) and v(x) = sin(100 x). Both have the same L^2 size (roughly), so an L^2 ruler calls them equally big. But v' = 100 cos(100 x) is a hundred times steeper, so integral of (v')^2 is about 10000 times bigger than integral of (u')^2. In the H^1 norm, v is enormously larger than u — the Sobolev norm punishes the rapid oscillation that L^2 ignores.
The H^1 norm feels oscillation through the gradient term; the L^2 norm is blind to it.
Different but equivalent formulas for the same Sobolev norm abound (sum of squares vs. square root, with or without cross terms); they give the same topology, so 'the' Sobolev norm is fixed only up to equivalence. The H^1 seminorm (gradient only) is a genuine norm just on subspaces like H^1_0 where constants are excluded — on all of H^1 it is only a seminorm, since a nonzero constant has zero gradient.