Positivity is a concept appearing quite frequently in the study of algebras and its related spectral theory. Positive elements naturally give rise to an ordering and therefore allows to construct special objects through an approximation (approximate identities, resolutions of the identity, etc.) Positive linear transformation then usually come next as special continuous linear operators (integrals, representations, etc.).
My question is what these are good for then apart from preserving the order.
Especially I'm wondering why some people even define an integral to be a positive linear transformation rather than just mentioning it as a side remark that the integral indeed happens to be positive. For now I can't enjoy positivity as much as continuity since I'm just missing its usefulness - and I hope your answers will change this so I can appreciate it the next time I encounter a positive linear transformation. :)
Physically, Hamiltonian operators in Quantum Mechanics should be semibounded, meaning that $(Ax,x) \ge M(x,x)$ for all $x\in\mathcal{D}(A)$ and for some fixed $M$. This has to be done with energy considerations. Second order ODES and PDES, in order to be symmetric, are quadratic in nature, and usually end up being semibounded--again, this is related to energy.
Mathematically, semibounded quadratic forms give rise to alternative inner-product structures on part or all of the underlying space, and this is the main tool for studying them. For this reason, quadratic forms which are not semibounded are difficult to study, and not generally as useful.
Reed and Simon Vol III has a lot about quadratic forms, especially semibounded ones, and especially in regard to Hamiltonian operators of Quantum.
$C^{0}$ semigroups associated with semibounded selfadjoint operators have interesting additional structure, too because they extend analytically to complex parameters in the semigroup: $$ U(z) = \int_{M}^{\infty}e^{iz\lambda}dE(\lambda),\;\; \Im z > 0. $$ Restricting $A=-i\frac{d}{dt}$ on $L^{2}(\mathbb{R})$ to its positive spectrum gives $$ LE[0,\infty)f = \int_{0}^{\infty}e^{i\lambda t}\lambda\hat{f}(\lambda)\,d\lambda $$ and suddenly you're looking at multiplication on the Hardy space of holomorphic functions on a half-plane. This is closely related to the Laplace transform, and much of this kind of structure extends to accretive operators on a Banach space. Accretive is a generalization of semibounded for a Banach space. There are few classes of operators on a Banach space for which such an extensive functional calculus exists. And this class of operators is useful in applications, too.
Pazy's classic book on $C^{0}$ Semigroups is a good, readable reference.