TITLE: What notions are used but not clearly defined in modern mathematics?
QUESTION [103 upvotes]: "Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
Felix Klein

What notions are used but not clearly defined in modern mathematics?

To clarify further what is the purpose of the question following is another quote by M. Emerton: 

"It is worth drawing out the idea that even in contemporary mathematics there are notions which (so far) escape rigorous definition, but which nevertheless have substantial mathematical content, and allow people to make computations and draw conclusions that are otherwise out of reach."

The question is about examples for such notions.
The question was asked by Kakaz

REPLY [9 votes]: The concept of turbulence is still vaguely or ill defined such as applied to too many phenomena. 
Examples from Is there a mathematically precise definition of turbulence for solutions of Navier-Stokes? and elsewhere

In the Ptolemaic Landau–Hopf theory turbulence is understood as a cascade of bifurcations from unstable equilibriums via periodic solutions ([the Hopf bifurcation][2]) to quasiperiodic solutions with arbitrarily large frequency basis.
According to [Arnold and Khesin][3], in the 1960's most specialists in PDEs regarded the lack of global existence and uniqueness theorems for solutions of the 3D Navier–Stokes equation as the explanation of turbulence.
Kolmogorov suggested to study minimal attractors of the Navier-Stokes equations and formulated several conjectures as plausible explanations of turbulence. The weakest one says that the maximum of the dimensions of minimal attractors of the Navier–Stokes equations grows along with the Reynolds number Re.
In 1970 Ruelle and Takens formulated the conjecture that turbulence is the appearance
of global attractors with sensitive dependence of motion on the initial conditions in the phase space of the Navier–Stokes equations ([link][4]). In spite of the vast popularity of their paper, even the existence of such attractors is still unknown. 
Existence of energy cascades (eg. Big vortices feeding on smaller vertices). This reflects the physical notion that mechanical energy injected into a fluid is generally on fairly large length and time scales, but this energy undergoes a “cascade” whereby it is transferred to successively smaller scales until it is ﬁnally dissipated (converted to thermal energy) on molecular scales
Von Karman: “Turbulence is an irregular motion which in general makes its appearance in ﬂuids, gaseous or liquid, when they ﬂow past solid surfaces or even when neighboring streams of the same ﬂuid ﬂow past or over one another.” 
Hinze: “Turbulent fluid motion is an irregular condition of the ﬂow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be discerned.”
Chapman: “Turbulence is any chaotic solution to the 3-D Navier–Stokes equations that is sensitive to initial data and which occurs as a result of successive instabilities of laminar ﬂows as a bifurcation parameter is increased through a succession of values.”
Criteria listed in McDonough's notes: 

nonrepeatability (i.e., sensitivity to initial conditions);
disorganized, chaotic, seemingly random behavior 
extremely large range of length and time scales
enhanced diﬀusion (mixing) and dissipation (both of which are mediated by viscosity at molecular scales)
three dimensionality, time dependence and rotationality (hence, potential ﬂow cannot be turbulent because it is by deﬁnition irrotational);
intermittency in both space and time.