It's important to remember that these signs/gestures are not just an idle formality to be used without comprehension. Each sign was originally shown to a student of the Golden Dawn as he/she underwent an elemental initiation ceremony. Signs from higher grades (courses of study) were not shown to students who had not been initiated into those grades. This was partly so that when initiation ceremonies were being conducted, it would be simple to make sure that no one was present who hadn't experienced an initiation into that grade yet.
The purpose of most of these signs is 3-fold: To act as a somatic/triggering device so as to help the practitioner achieve a previous state of consciousness (that of the original elemental initiation ceremonies in which he/she first learned the signs/gestures), and (in the case of the Neophyte Signs) to act as focusing tools for the actual manipulation of energy and consciousness. For example, it's easier to effectively open an astral 'portal' to an elemental realm when you're also engaging your hands to open a portal in the air in front of you.
A final use of these signs is often for purposes of testing and greeting entities. If you make the sign of Earth at an entity and it isn't able to return that sign to you, then supposedly the vision you've entered is false. The LVX signs are generally used to greet and test spiritual/angelic/divine beings. There is some debate over the effectiveness of using such signs for testing of entities.
The Sign of Horus - Take a step with your right foot and thrust the hands forward as you forcefully vibrate a name. This sign is normally used as an aid in charging sigils drawn in the air. Can be used in the Lesser Ritual of the Pentagram to charge the drawn Pentagrams with energy.
The Sign of Harpocrates - This sign almost Always follows the Sign of Horus to cut off the energy current that was being sent forth while charging sigils etc. Simply return to a normal standing position, and point the forefinger of the Left hand at the lips (as in Egyptian art depicting the god Harpocrates- Horus the child, a silent god).
To form the Sign of Osiris Slain, simply extend your arms straight out to both sides as if you were forming the letter 'T' with your body.
For the Sign of the Morning of Isis, there are two versions you may choose from. The original version is created by simply putting your right arm up at a 90 degree angle to your left arm which is straight out to your left side (where it was in the previous sign), thus forming a letter 'L' with your arms. To create the second version of this sign instead (if you prefer), then you'll have to look into the thelemic works of Crowley. It's too difficult to describe the sign here without a picture. His version of this sign is one of the few things by him that I don't mind.
To form the Sign of Apophis and Typhon, raise the arms at an angle of 60 degrees to each other above the head, which is thrown back.
To form the Sign of Osiris Risen, cross the arms on the breast and bow the head.
The Rending of the Veil (no picture available)- This sign is more of an action than a living symbol. To perform, stand normally and extend the hands in front of you with palms turned outward as if to spread open a veil or curtain. Then, part the imaginary curtain (make the motion in front of you). Spread the imaginary curtain wide enough to walk through, but don't step forward (unless the working requires it). This sign is usually performed when one wishes to open a link or portal to another plain. Sometimes called the Active Sign of the Portal.
The Closing of the Veil (no picture available)- This sign is also more of an action than a living symbol, and is used together with the Rending of the Veil. To perform, extend the hands in front of you with palms facing each other as if to close a curtain that was opened wide enough to walk through. Then, close the imaginary curtain (make the motion in front of you). This sign is usually performed when one wishes to close a link or portal to another plain. Sometimes called the Passive Sign of the Portal. In a working, you would use the Rending of the Veil to open a portal, and when you're done working with that portal you would use the Closing of the Veil to seal it back up. Both Signs should feel very natural to perform, as they are motions similar to what you would use if you needed to enter or exit a room that used hanging beads or curtains for a door.
Picture temporarily borrowed from Israel Regardie's "The Golden Dawn" by Llewellyn. I'll try to update it with an original picture soon.
Zelator (1=10) refers to Earth
Theoricus (2=9) refers to Air
Practicus (3=8) refers to Water
Philosophus (4=7) refers to Fire
Above are the elemental grade signs as used by the Golden Dawn. Water and Fire signs are almost identical in hand positioning, only the arm positions have changed. These are the signs used in the Supreme Invoking Ritual of the Pentagram (which makes some use of Enochian forces).