As I understand it, generally, a higher gain antenna will result in a longer range and better reception for FPV flying. What is antenna gain? What makes a low-gain patch antenna different that a high-gain patch antenna?
Gain, as a general term, is a measure of how large or strong something (a voltage, a radio wave, a sound wave, or whatever else) is relative to some baseline value of the thing being measured. A gain is a ratio: If a device takes a voltage on one end and makes it 20x larger (amplifies) it 20-fold, it has a gain of 20. In practice they are represented using decibels, a special unit that makes it easy to work with very large or small ratios. See the note at the end of this answer for more information about them.
Consequently, the gain of an antenna is the ratio of how much stronger the radio waves generated by the antenna are, relative to those generated by a standard "reference antenna" when fed by the same signal. This reference antenna used as a baseline is considered to have a gain of 0 dB, or 1:1 (equal to itself).
The reference antenna is usually one of two types: An ideal isotropic antenna, which radiates equally in all directions with zero losses, or a dipole antenna. To indicate which one is used as the baseline, a letter is added to the end: dBi represents gain relative to an isotropic antenna, and dBd relative to a dipole.
The dipole itself has a gain of 2.15 dBi (and 0 dBd), so to convert a dBi gain to a dBd gain you subtract 2.15 from it, or add the same number to convert the other way around.
Every antenna receives the same way as it transmits, so an antenna's reception gain (how strong of an electric signal is generated in it by a radio wave, relative to the chosen reference antenna) is always the same as its transmission gain.
An antenna with 3 dB (2x) gain in a certain direction will transmit 2x the power in that direction, but also radio waves coming from this direction will generate 2x stronger signals for a connected receiver to detect.
If an antenna's gain is 13 dBd (a 20:1 ratio), its signal will be 20 times as strong as a dipole's (and its reception will be 20x stronger as well)! Wonderful, right?
Well, there is a caveat to high gain, and it basically boils down to: There ain't no such thing as a free lunch.
Gain and directionality
True isotropic antennae do not exist in the real world, i.e. no real antenna radiates equally strong in all directions. There are directions in which it will radiate the strongest, and others in which it doesn't radiate at all. So the gain of a real antenna is not the same in all directions either: in some it is stronger, in some it is weaker. The easiest way to show this is a graph showing the antenna’s gain in every direction around the antenna. That graph is called a radiation pattern and looks like this:
The bold line represents the gain: the further it is from the center, the bigger the gain. Above is the radiation pattern of a directional antenna: the gain in one direction is a lot more than in other directions.
So when we speak of an antenna's gain, we really mean its gain in a certain direction. And that direction is usually the one in which the antenna's gain is the strongest.
Also, you can't get more out of an antenna than you put into it. The sum of all power radiated in all directions is always equal to the power that goes in (Minus a small percentage of losses); Antennae create gain by radiating this power such that the created waves add up in the preferred direction and cancel in others.
So if an antenna has a higher than baseline (positive) gain in some direction, you pay for that by having a negative gain in other directions. Gain only focuses power — it does not create power.
For our imaginary 13 dBd antenna it means that only a small sector will get this much gain, and all the other directions will have almost zero radiated power to make up for it.
So gain is not only the measure of how well an antenna receives or transmits, it's also a measure of the antenna's directivity: If the antenna focuses all of its power in a tight cone, it will have a huge gain within that cone, and a huge gain rating on the box... But its gain outside of that cone will be almost nonexistent. So when you are choosing a directional antenna, the gain is also an indication how directional it really is. Lower-gain patch antennae will give you a soft, forgiving cone but lower gain in it, while high-gain directionals will have a very tight zone in which they could transmit or receive at all, but within it they will be super powerful.
Omnidirectional antennae (which, as you now know, are never truly omnidirectional) have a radiation pattern that's more or less shaped like a donut:
They have an equal (and positive) gain in any direction perpendicular to the antenna, which gets lower as the direction gets closer to parallel to the antenna. These antennae also have their own kind of directivity: if you mount the antenna vertically, the "horizontal" gain will be the same in any direction, but it can be higher or lower relative to the "more vertical" directions, making the donut flatter. In other words, the antenna can prioritize gain towards the horizon versus overhead gain more or less.
A note on decibels
In many engineering fields ratios can easily get extremely large or extremely minuscule and tend to multiply a lot. Radio is one of the most prominent of those. At the input to a receiver, signals are frequently smaller than one ten-billionth of a watt. When they come out of a transmitter, they’re often measured in kilowatts! Antennas, propagation and electronic circuits change signal strengths by many factors of ten.
To manage these vast differences in value, engineers measure ratios in decibels, or dB, which represent the ratio of two quantities as a power of 10. The formula for computing decibels is: dB = 10 * log_10 (ratio)
This is a logarithmic unit, meaning that when two ratios are multiplied, their representations in dB are simply added up: 10 decibels represent a ratio of 10:1; 20 dB is 1010=100:1, and 30dB is 1010*10=1000:1.
Positive values of dB mean the ratio is greater than 1 and negative values of dB indicate a ratio of less than 1: -10 dB represents a ratio of 1/10, and 40dB is 1/10000.
Here's a quick cheat sheet of dB to actual ratio for numbers smaller than 10:
1dB corresponds to about 1.25:1
3dB is almost exactly 2:1 (this is the most important one to remember)
7dB ~= 5:1
10dB is 10:1 (exactly)
For everything else, you just add up the numbers you know:
6dB (=3dB+3dB) is ~ 4(=2*2):1, and 9dB is ~ 8:1
13 dB is 10 dB + 3 dB, so 10*2=20:1.
46 dB is 40 dB + 6 dB = 10^4 * 4 = 40000:1
15 dB is 10dB + 5dB; and 5 dB is somewhere between 3dB and 6dB (2 and 4), so it's probably three point something. Thus our guess would be 15dB is somewhere in the 30-35 to 1 range.
'Antenna gain' gives you the power transmitted by an antenna in a specific direction, when compared to a theoretical antenna that radiates equally in all directions (this is known as an 'isotropic antenna')
In effect, 'Antenna gain' defines how strong a signal a particular antenna can transmit or receive in a specified direction.
Antenna Gain is related to the directivity of the antenna, but also takes into account other losses in the system (for example, surface coating irregularities, resistance, manufacturing faults, dielectric, etc.).
So, in essence:
Antenna gain = Directivity x Efficiency
Although the actual formulae are a little more complicated!
Antenna gain is a measure of how effectively an antenna sends or receives a signal. It is a relative measurement and is usually a comparison to a simple antenna design called a dipole antenna (dBd) or an ideal reference (dBi) - often a manufacurer picks whichever of these numbers happens to be bigger.
Gain of below 1 means the performance is worse than this reference level, but there can be advantages to this such as the antenna being smaller (which in a drone context, means it weighs less and is shorter - so easier to position and keep out of the propellers!)
The gain can also be directional. It may help to think of a torch - the bulb is omnidirectional but you need light to go forwards, so you have a shiny reflector behind the bulb to give you maximum brightness ('gain') in one direction. A 'radiation pattern' can be produced to show the gain on an antenna in different directions.