What is meant by magnetic circulation in ampere's circuital law? So i get what the basic formula means, but what does the term magnetic field times length signify, what is exactly meant by magnetic circulation and how was Ampere able to come up with this?
If you could explain with an analogy that would be great.
 A: Short answer:
An electric current (or a time-varying electric field) in space will generate a magnetic field that appears to "flow", or circulate around the current (or time-varying electric field) similar to how water flows around the axis of a vortex.
Long answer:
Consider an orientable piecewise-smooth 2-dimensional surface $\Sigma$, with a boundary defined by a simple, closed, piecewise-smooth 1-dimensional curve $\partial \Sigma$, embedded in a region of 3-D space where either a current density (current per cross-section area) $\mathbf J$, or a time-varying electric field $\mathbf E$ pass through the surface. Then a magnetic field $\mathbf B$ will be induced in that region of space such that,
$$\oint_{\partial \Sigma} \mathbf B \cdot d\mathbf l = \mu_0 \iint_{\Sigma} \left(\mathbf J + \epsilon_0\frac{\partial \mathbf E}{\partial t}\right) \cdot d\mathbf S$$
Applying the Kelvin-Stokes' Theorem, we can rewrite this relationship as
$$\nabla \times \mathbf B = \mu_0 \mathbf J + \mu_0\epsilon_0\frac{\partial \mathbf E}{\partial t}$$
These equations are referred to as, respectively, the integral and differential form of Ampere's Law with Maxwell's correction. Ampere came up with this (and Maxwell later corrected it) after observing numerous experiments that suggested that this law is true.
The form of Ampere's Law is analogous to how the behavior of vortices is described in classical fluid dynamics. Often, physicists will described the behavior of a fluid containing vortices using the concept of circulation, denoted by $\Gamma$ and defined as,
$$\Gamma = \oint_{\partial \Sigma} \mathbf v \cdot d\mathbf l = \iint_{\Sigma} \mathbf \omega \cdot d\mathbf S$$
where $\mathbf v$ is the velocity flow field of the fluid and
$$\mathbf \omega = \nabla \times \mathbf v$$
is called the vorticity of the flow.
If you are unfamiliar with any of these concepts from multivariable calculus -- specifically the concept of the curl of a vector field $\mathbf F$, denoted as $\nabla \times \mathbf F$ or sometimes as $\mathrm{curl}\left(\mathbf F\right)$ -- then I suggest you watch this series of videos, as well as the other videos on multivariable calculus, put out by Khan Academy with guest lecturer Grant Sanderson of the YouTube channel 3blue1brown, as they should at least give you a decent grasp of the intuition behind the math:
2d curl intuition
2d curl formula
2d curl nuance
3d curl intuition, part 1
3d curl intuition, part 2
3d curl formula, part 1
3d curl formula, part 2
