Functions" are generally what define the organization chart. Accounting. Facilities. R&D. Manufacturing. Marketing. Sales.
Usually a cross-functional team or process means a number these groups must work together to support, facility or operate part of the corporate value chain and thus "needing to work together".
For example, marketing working accounting and R&D to resolve a product issue in the value chain, would be a cross-functional activity. Typically a "team" is a project team assembled for a short-term issue. A process is an ongoing collaboration between functions.
Because of the inherent tendency of functional groups in a hierarchy to "silo themselves" and live in their own worlds, operating "cross functionally" is difficult for pretty much all businesses.
This is why it's so often necessary to create "cross-functional teams" to deal with real world business problems (i.e. stuff that makes money which is always the value chain). Keeping such a team together over the long term is often challenging. Thus if you can create an inanimate process to either perform the function they once performance as a special instance or to trigger the assembly of a new team if required, you'll usually do that instead.
Functional groups seem stupid in this context because they are often "the problem" with most organizations - why are they separate in the first place if their cooperation and collaboration is so critical. However, each function typically involves a concentration of skills, focus, talent and personality types which would always work poorly with other different groups.
Also the hierarchy is the "official" means to delegate resource allocation from the board of directors down to executive and then down to managers and line employees: the need to intelligently manage that delegation of the most precision company powers trumps the value chain for survival reasons.
Occasionally different organization structures are used to address this. For example the "matrix organization" is one common alternative to the hierarchy. Unfortunately it requires a fairly high skill and social quotient level to work well but in some cases a matrix is a good fit to deal with cross-functional gaps.