Lens used for glasses/contact lens and cameras This has been confusing me. Let's take convex lens. Below in the picture, is it a double convex lens or just the symbol of convex lens?

For example: In convex glasses and contact lenses, the lens used is only half of what is shown above.
Upon some further searching I found this picture:

So, does it mean the lens we use glasses is Plano-convex lens? and the lens we use for cameras and magnifying glasses are double convex lenses?
 A: If one wants to be rigorous then the lens in the first picture should be called double convex or bi-convex since it is convex on both sides. 
For camera lenses and magnifying glasses you mostly want to enlarge the picture and use convex lenses.
For the glasses you have to distinguish if the person is far- or nearsighted. For farsighted people you would take convex lenses, for nearsighted people you would take concave lenses. This is related to the length of the eyeballs and less to magnifying the picture.
Edit:
In the first picture of your question the rays on the left side of the biconvex lens seem to run parallel to each other. This representation is often misleading since in reality the rays would converge and focus at one point (even if the point is far away.) For example look at this plan-convex lens: 

In a plan-convex lens the rays on the left are coming parallel and may focus only at infinity. You can always construct such pictures yourself just remembering that the angles of rays with respect to the normal are larger in the medium with lower density (air). See for example the red ray in the picture above. By drawing the same picture for a bi-convex lens you would see that the rays would not remain parallel on the left side after leaving the lens, they would converge and focus in a single point similar to the right side of the lens. 
In reality this is what you want to achieve for your farsighted glasses, magnifying glasses and so on. So you would use bi-convex lenses although the bending on both sides might be different. For nearsighted glasses you would rather use bi-concave lenses. Of course there are applications where plan-convex lenses would be preferred, e. g. to reduce unwanted optical effects (in telescopes and so on... see this question here).
A: There seems to be a confusion between two different aspects of a lens.
A converging lens takes a parallel beam of light and brings it to a focus at a point. It is generally thicker at the centre than at the edge.  Examples of this are:  a magnifying glass, and eyeglasses for an individual who is farsighted and cannot focus on nearby objects.
A diverging lens takes a parallel beam of light and spreads it out, as if it were coming from a point. Such a lens could be used for glasses for a near-sighted person who cannot focus on distant objects
From http://www.sciencefacts.net/diverging-lens.html

None of the above says anything definitive about the shape of the each of the two surfaces of the lens; only about the relative thickness at the centre and edge. 
So:


*

*Convex-convex - always converging; always thicker at the centre

*Plano-convex - always converging; always thicker at the centre

*Concavo-concave - always diverging; always thinner at the centre

*Plano-concave - always diverging; always thinner at the centre

*Convex-concave - can be converging, diverging or neutral


All these lenses will behave the same way, no matter which way the light goes through them.
Eyeglass lenses will almost always be convex on the outer surface, the one farthest from the eye, simply to fit it to the curvature of the face.
If the inner surface is concave, and more sharply curved than the outer, then the lens is diverging.
If the inner surface is more gently concave, flat, or convex, then the lens is a converging one.
Stage glasses, as part of the costume for an actor with normal vision, would have a convex outer surface and a concave inner surface, with the same radius of curvature for both surfaces, and would be neither converging or diverging...
A: Spectacle lenses for short sight are essentially biconcave, although the whole lens is curved making it concave (inner surface) / convex (outer surface).  For long sight they will be biconvex but again curved making them concave / convex (but less concave than lenses for short sight).
Camera lenses have multiple elements and are complicated to describe: modern camera lenses are the highly-optimised result of two hundred years of evolution and (more significantly) extensive computer-optimised design: zooms in particular hardly existed before numerical simulation techniques.

Yes, I know there were zoom lenses before computer design: the reason people still worship primes is largely because of how rubbish such lenses were, not to mention how big.  The tiny zoom on your P&S camera with its huge range is entirely a result of computer design.
