What is the actual significance of the focal point of a lens?

Look at the image above where the three laws of optics are shown:

1. Beam passing through the optical centre passes untouched.
2. Beam passing parallel to the optical axis leaves through (projective) focus.
3. Beam passing through (objective) focus leaves parallel to the optical axis.

Using these three laws you can construct how an object will be projected.

It is easy to see that the focal length $f$ and the object position $S_1$ describes:

1. Magnification $h_2/h_1$,
2. Position of the projection $S_2$.

When thinking about the effect of lenses used in binoculars, one must take eye lens into account. And here it is quite tricky.

The focal length tell us how much the light rays will be bent. Think of the light rays as a paper cone, just like the one you get when you buy a snow cone. The mouth of the cone is the size of the lens, and the point of the cone is the focal point. The length of the cone is the focal length. Now picture a cone where the point is very close to the mouth. It would be a very steep short cone. The light rays would come in at very steep angles, and after they crossed the focal point they would spread out quickly. In fact, they would make another cone leaving the focal point that matches the cone that entered the focal point. The rays would continue on spreading out wider and wider. Now picture a cone that is very long, say three feet, a novelty snow cone, notice how these rays come into focus at a much smaller angle, and after they pass the focal point they will form another long cone on there way out. Well, there you have it, the focal length tells us how long the snow cone is. I hope your not offended by my simple examples and language in this answer. It's how I think of focal length. With a little imagination you can see how different cone sizes would be applicable to different applications.

We're struggling with a conceptual problem here. We don't need math for this one. The original poster is confusing the object he's looking at with the light rays bouncing off of (or emitting from) that object. I just went down this rabbit hole myself.

Our eyes have lenses and our retinas serve as the focal plane. No different from a camera or a telescope or whatever. Calling it a focal "point" is misleading insofar that our entire field of view resolves across the entire surface of our retina, not on one discreet point. A focal "point" should be thought of as a focal "plane" some specific distance from the lens. Not a single dot in space.

If you look at a tiny little point of light, perhaps a power button indicator on your PC or something. That tiny little point of light is emitting photons in all sorts of directions at once. The only photons from that tiny point of light your eye receives are the ones that happen to reach the surface of the lens of your eye. But the ENTIRE lens of your eye is receiving photons from that one tiny discreet point of light. If your eye didn't have a lens at all then that single point of light would resolve into a blanket of whitish opaqueness. (A "blur")

It's the lens of your eye that is bending all the light rays coming off that tiny crisp point of light that now blanket the entire surface of your eyeball lens back into that tiny crisp point of light representative of where the light originated from in the first place. It resolves back to that tiny point of light right at the surface of your retina where the information can then be sent along to your brain. And that single lens is performing this trick for your entire field of view and resolving your entire field of view into a beautiful crisp rendering over the entire surface of your retina.

Light scatters in all directions. A lens focuses scattered light rays back into something visually representative of their point of origin.

The lines you see entering lenses in diagrams don't typically represent the object, they represent the light rays coming FROM the object. Two different things.

Also, for a convex lens. If I have my eye at the focal point, would the image appear tiny? Since all the light rays converge there?

I think the error in this line of thinking should be apparent now. All the light rays coming into the optical system aren't converging to a point. They're focusing onto a plane. Only the light rays from a single tiny point of origin (like a power button or a star) are going to converge back to a tiny point. As for what you would actually see, it's important to remember that the optical system of your eye becomes an extension of the mechanical optical system you're looking through. The only focal plane that REALLY matters is the back of your retina.

Not to evade the question I guess. What would you see? I'm not entirely sure to be honest. I suspect you'd see a nice crisp rendering of whatever it is you're looking at. Much like looking at the surface of a flat screen television display.

To answer this question, oneself or a group of learners needs to understand what is focal length and its importance. Thus, that it will give more clear insight if a learner knows how to differentiate and how to draw the diagrams of each lens like concave (diverging lens) and convex (converging lens).

However, lets define focal length. Focal length is the distance from the lens (vertex) to the focal point. The focal point is the point which enables light rays to parallel the axis. The significance (important) of a lens to have a focal length is that;

1. It determines the image location after the object being reflected or refracted from the lens.

2. It shows the magnification of an image form from an object, whether it enlarges or diminished.

3. And also, it determines image height. This can be seen when the object was moved closer or further from or to the lens. Sometimes there is no image form when the object was placed on the focal point and there are times when the image form behind the mirror as the object was continually moved towards the lens.