Main Difference
The main difference between Circle and Ellipse is that the Circle is a simple curve of Euclidean geometry and Ellipse is a type of curve on a plane.

Circle
A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term “circle” may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.
A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the twodimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.

Ellipse
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a special type of an ellipse having both focal points at the same location. The shape of an ellipse (how “elongated” it is) is represented by its eccentricity, which for an ellipse can be any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.
Ellipses are the closed type of conic section: a plane curve resulting from the intersection of a cone by a plane (see figure to the right). Ellipses have many similarities with the other two forms of conic sections: parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder.
Analytically, an ellipse may also be defined as the set of points such that the ratio of the distance of each point on the curve from a given point (called a focus or focal point) to the distance from that same point on the curve to a given line (called the directrix) is a constant. This ratio is the abovementioned eccentricity of the ellipse.
An ellipse may also be defined analytically as the set of points for each of which the sum of its distances to two foci is a fixed number.
Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency. A similar effect leads to elliptical polarization of light in optics.
The name, ἔλλειψις (élleipsis, “omission”), was given by Apollonius of Perga in his Conics, emphasizing the connection of the curve with “application of areas”.

Circle (noun)
A twodimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from a given point (center).
“The set of all points (x, y) such that (x1)^{2} + y^{2}“
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“r^{2} is a circle of radius r around the point (1, 0).”

Circle (noun)
A twodimensional geometric figure, a disk, consisting of the set of all those points of a plane at a distance less than or equal to a fixed distance (radius) from a given point.

Circle (noun)
Any thin threedimensional equivalent of the geometric figures.
“Put on your duncecap and sit down on that circle.”

Circle (noun)
A curve that more or less forms part or all of a circle.
“move in a circle”

Circle (noun)
Orbit.

Circle (noun)
A specific group of persons; especially one who shares a common interest.
“inner circle;”
“circle of friends”

Circle (noun)
A line comprising two semicircles of 30 yards radius centred on the wickets joined by straight lines parallel to the pitch used to enforce field restrictions in a oneday match.

Circle (noun)
A ritual circle that is cast three times deosil and closes three times widdershins either in the air with a wand or literally with stones or other items used for worship.

Circle (noun)
A traffic circle or roundabout.

Circle (noun)
Compass; circuit; enclosure.

Circle (noun)
An instrument of observation, whose graduated limb consists of an entire circle. When fixed to a wall in an observatory, it is called a mural circle; when mounted with a telescope on an axis and in Y’s, in the plane of the meridian, a meridian or transit circle; when involving the principle of reflection, like the sextant, a reflecting circle; and when that of repeating an angle several times continuously along the graduated limb, a repeating circle.

Circle (noun)
A series ending where it begins, and repeating itself.

Circle (noun)
A form of argument in which two or more unproved statements are used to prove each other; inconclusive reasoning.

Circle (noun)
Indirect form of words; circumlocution.

Circle (noun)
A territorial division or district.
“The ten Circles of the Holy Roman Empire were those principalities or provinces which had seats in the German Diet.”

Circle (noun)
A bagginess of the skin below the eyes from lack of sleep.
“”After working all night, she had circles under her eyes.”

Circle (verb)
To travel around along a curved path.

Circle (verb)
To surround.

Circle (verb)
To place or mark a circle around.
“Circle the jobs that you are interested in applying for.”

Circle (verb)
To travel in circles.
“Vultures circled overhead.”

Ellipse (noun)
A foci of the ellipse) is constant; equivalently, the conic section that is the intersection of a cone with a plane that does not intersect the base of the cone.

Ellipse (verb)
To remove from a phrase a word which is grammatically needed, but which is clearly understood without having to be stated.
“In B’s response to A’s question: (A: Would you like to go out?, B: I’d love to), the words that are ellipsed are go out.“

Circle (noun)
a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the centre)
“draw a circle with a compass”

Circle (noun)
something in the shape of a circle
“the lamp spread a circle of light”
“they all sat round in a circle”

Circle (noun)
a dark circular mark below each eye caused by illness or tiredness
“she was pale and rather beautiful, with dark circles around deep, exhausted eyes”

Circle (noun)
a curved upper tier of seats in a theatre or cinema
“she sat in the front row of the circle”

Circle (noun)
short for striking circle

Circle (noun)
a group of people with a shared profession, interests, or acquaintances
“she did not normally move in such exalted circles”

Circle (verb)
move all the way around (someone or something), especially more than once
“they were circling Athens airport”
“we circled round the island”

Circle (verb)
move in a wide loop back towards one’s starting point
“he paced away from her, then circled back”

Circle (verb)
form a ring around
“the abbey was circled by a huge wall”

Circle (verb)
draw a line around
“circle the correct answers”