An equatorial sundial is a sundial which takes advantage of the regularity of the movement of the sun through the sky during the (solar) day which is most apparent at the Equator - the time of day is directly linked to the angle between the sun and a line in the sky (or a plane intersecting the celestial sphere) containing the points (directions) zenith and due North. The time is read from a dial which is parallel to the equator, on which every hour is an equal size (angle).
There are two types of equatorial sundial. One type uses a disc, the other type uses half a cylinder.
A disc-type equatorial sundial uses a disc with a straight gnomon which extends normally (or perpendicularly, at a right angle) from the disc on both sides. The disc is fixed to the ground so that it is parallel to the equator, and the gnomon points North-South (i.e. the gnomon is parallel to the axis of rotation of the Earth). This means the angle between the disc and vertical is equal to the latitude (or, the angle between the disc and horizontal is equal to the colatitude, or, the angle between the gnomon and horizontal is equal to the latitude). On the equator, the disc is placed vertically, the gnomon points horizontally (to due North on the horizon and due South on the horizon). In the Northern hemisphere, the gnomon is pointing into the sky to the North at one end and into the ground to the South at the other. In the Southern hemisphere, the gnomon points up and South on one side and down and North on the other. Once the disc is in place, mark noon at the lowest point on the disc, and mark each of the 24 hours equally spaced around the circumference.
When it's Summertime in the Northern hemisphere, the North face of the disc is used to tell the time (the South face is in shadow) - no matter where on Earth the sundial is. When it's Summertime in the Southern hemisphere, the South face of the disc is used to tell the time. Near the equinoxes, the sundial is useless - the sun hits the disc edge-on, the shadow of the gnomon is either very faint or nonexitstent, and near the solstices is the easiest time to read the sundial. In order that the shadow of the gnomon extends to the edge of the disc, the angle made from one end of the gnomon to the edge of the disc the other end of the gnomon (assuming it extends the same distance on either side of the disc) must be at least twice the axial inclination of the Earth (twice 23.5° = 47°). Put another way, the ratio of the radius of the disc to the length of the gnomon on either side must be tan(23.5°), or, the length of the gnomon must be at least 0.44 times the radius of the disc (on each side).
It is worth noting that even though every day on the Equator has exactly 12 hours of sunlight, from precisely 6am to 6pm (sundial time), the equatorial sundial at other latitudes can show other times. In Summer, when there are more sunlight hours, the upper sundial will show sunrise at earler than 6am and sunset at later than 6pm. (The lower sundial will only work in the Winter half of the year, when there are 12 hours or fewer of sunlight.) This sundial even works in the Arctic and Antarctic circles, where the sun stays above the horizon for a full day (or longer).
The other type of equatorial sundial uses half a (hollow) cylinder. The cylinder is cut along a plane parallel to the axis, i.e. instead of ending up with two halves each with a full circle at the end, you end up with two halves and each of them has two semicircles, one at each end. Stand a cylinder on a table, and imagine cutting it vertically instead of horizontally.
The inside curved surface of the semicylinder is marked with 11 lines, each one parallel to the axis of the cylinder, dividing the surface into 12 equal parts. A string is placed at the axis of the cylinder, tied from the middle of the semicircle at one end to the middle of the semicircle at the other end (actually not the middle of the semicircle, the middle of the circle that the semicircle came from). The semicylinder is positioned so its gnomon is parallel to the Earth's axis (just like the gnomon of the disc-type equatorial sundial), and so that it faces to the sky (the gnomon must be above the dial, and the straight lines of the semicircles must be horizontal).
The lowest line inside the semicylinder represents noon, the edges represent 6am and 6pm. This sundial has the disadvantage that it does not show the extra hours of daylight in Summertime. That disadvantage could be overcome in one of two ways; either the material of the semicylinder could be transparent enough that sunlight gets through it but opaque enough that the shadow of the gnomon is still visible on it, and use more than half of a cylinder (inside the Arctic and Antarctic circles, use the full cylinder); or, the semicylinder could rotate about its axis a bit (up to quarter of a revolution in either direction) - making sure that one of the lines is the lowest it can be, hours can be counted as lines from noon (the lowest) to earlier than 6am or later than 6pm.
In order for this sundial to work all year (particularly at the solstices), the semicylinder must be tall enough (relative to its radius) that the shadow of the gnomon will always hit someplace on the curved surface. In other words, the angle made by one end of the 'noon' line to one end of the gnomon to the other end of the 'noon' line must be greater than the axial inclination of the Earth. The ratio of the height (or length) of the cylinder to the radius must be at least tan(23.5°), or, the height of the cylinder must be at least 0.44 times the radius. A squat cylinder will only work near the equinoxes. (The semicylinder works opposite the disc in that the semicylinder is easier to read nearer the equinoxes, and harder to read nearer the solstices.) Generally, the taller the cylinder, the easier it will be to read anytime.
These sundials are appealing, because they illustrate (better than other sundials) that the Earth spins. The hours are all equal angles because the rate of spin is constant, the gnomon is parallel to the axis of the spin. They bring the simplicity of measuring time at the equator to any latitude, and (if one of the fixes for the hemicylinder sundial is used) they fully work even though circumstances are different than at the equator. Even apart from the seasonal extra hours of sunlight, these instruments work to full capacity for example at a mountain peak where the sun may be visible below horizontal. They are elegant, both in terms of engineering and user interface.