A curved line forming a closed loop, where the sum of the distances from two points (foci) to every point on the line is constant.

Try this Drag any orange dot. You can change the position of the two focus points (F1, F2).
Also drag the point on the ellipse and observe that the sum of the lengths of the lines that meet there is constant.

An ellipse looks like a circle that has been squashed into an oval. Like a circle, an ellipse is a type of line. Imagine a straight
line segment that is bent around until its ends join.
Then shape that loop until it is an ellipse - a sort of 'squashed circle' like the one above.
Things that are in the shape of an ellipse are said to be 'elliptical'.

How ellipses are defined

An ellipse is defined by two points, each called a focus. (F1, F2 above).
If you take any point on the ellipse, the sum of the distances to the focus points is constant.
In the figure above, drag the point on the ellipse around and see that
while the distances to the focus points vary, their sum is constant.
The size of the ellipse is determined by the sum of these two distances.
The sum of these distances is equal to the length of the
major axis (the longest diameter of the ellipse).

The two lines a and b that define the ellipse are called
generator lines. Each one is sometimes called a
generatrix.

The position of the foci (plural of focus, pronounced 'foe-sigh') determine how 'squashed' the ellipse is.
Drag F1 and F2 and see how this happens.
If they are at the same location, the ellipse is a circle. A circle is, in fact, a special case of an ellipse.
In the figure above, drag one focus until it is over the other.

Properties of an ellipse

Center

A point inside the ellipse which is the midpoint of the line segment linking the two foci.
The intersection of the major and minor axes.

Major / minor axis

The longest and shortest diameters of an ellipse.
See Major / Minor Axis of an Ellipse.
The length of the major axis is equal to the sum of the two
generator lines (a and b in the diagram above).

A line segment linking any two points on an ellipse.

Tangent

A line passing an ellipse and touching it at just one point.
See Tangent to an Ellipse

Secant

A line that intersects an ellipse at two points.

Relation to a circle

A circle is actually a special case of an ellipse.
In an ellipse, if you make the major and minor axis the same length, the result is a circle, with both foci at the center.
See Circle definition

Using Trigonometry:
where
t is the parameter
a is the horizontal semi-axis and
b the vertical semi-axis.
For more see Parametric equations of an ellipse.

Using the formula
When the center of the ellipse is at the origin (0,0):
where
a is the horizontal semi-axis and b the vertical semi-axis
(x,y) are the coordinates of any point on the ellipse.
For more see General equations of an ellipse.

Using the formula
When the center of the ellipse is at the point (h,k):
where
a is the horizontal semi-axis and b the vertical semi-axis,
(h,k) are the x,y coordinates of the center of the ellipse
and
(x,y) are the coordinates of any point on the ellipse.
For more see General equations of an ellipse.

As a conic section
When an ellipse is cut at an angle by a plane, the intersection is in the shape of an ellipse.
For more see Conic section - ellipse.