I may be able to offer some hints. I am also looking forward to more answers (Ryan please!) because this is definitely a complicated subject, with contributions from people coming from very different backgrounds (hep-th, cond-mat, pure math, etc.).
"Condense" may mean different things to different people. I believe the oldest precise definition was that an operator condenses if it picks up a vacuum expectation value. The physical interpretation is that the operator acquires a macroscopic value, i.e., it becomes something like a constant background, a spacetime filling object.
Note that vevs typically imply that a symmetry is spontaneously broken (and not the other way around!). Indeed, if the operator is charged under a given symmetry, then its condensate breaks the symmetry. Of course there are situations where an operator that is not charged under any symmetry condenses, and in those cases the vev does not break any symmetries, and thus there is no spontaneous symmetry breaking. The simplest example is, I think, the Schwinger model, where the basic meson condenses but there are no symmetries to begin with, so nothing is broken. Of course, Schwinger is gapped and has a unique, trivial vacuum state.
In QCD, the object that typically condenses is the quark bilinear $\psi_L\psi_R$. If this operator has a non-zero expectation value, it breaks the chiral symmetry $SU(N)_L\times SU(N)_R$ down to the diagonal subgroup. This breaking means that there are Goldstone bosons, the pions. There are other objects that could condense beyond the quark meson (e.g., with enough supersymmetry it is known that certain monopoles condense; I believe they break the magnetic $U(1)$ one-form symmetry and hence the Goldstone photon becomes massive and is gapped out, hence the theory becomes confining. But don't quote me on that).
More generally, condensing means that you allow the object to have a macroscopic value. In cond-mat, I believe, this is often achieved, roughly speaking, by dropping certain terms from the Hamiltonian, i.e., those terms that energetically penalize the excitation of the object you want to condense. If the object to condense does not have a term in the Hamiltonian, then it will cost no energy to produce it, and the vacuum state will have a non-zero excitation number for it, i.e., the object has acquired a macroscopic value.
A different notion of condensing is that of gauging. If you gauge a symmetry generated by that operator, then effectively you allow that operator to proliferate, because gauging mean that you identify any two objects that differ by the operator you gauged. Thus, the gauged operator is identified with the vacuum. This is reminiscent of the previous paragraph, since the Hamiltonian must be gauge invariant, i.e., it cannot depend on the object you gauged. (Note that in cond-mat, gauging is often enforced energetically, i.e., you impose the Gauss law by penalizing configurations that violate it).
This last notion is the one used for strings. Condensing a string means gauging the one-form symmetry it generates. Gauging means summing over bundles, i.e., you introduce a mesh of strings, which reproduces the fact that the condensed object acquires a macroscopic value. When you gauge a string, you sum over all possible insertions, and therefore the string appears everywhere: the vacuum consists of a spacetime filling mesh of strings.