If a laptop is always plugged in and charging, what is the effect on the battery?

It will not damage your battery. 

It will shorten its life though, but there is nothing practical you can do about it.

Lithium Ion batteries age in two ways, one is through what is termed "Calendar Life" and the other is Cycle life.  Here is what you need to understand about both:

Calendar life  - the time spent at a given state of charge (SOC) and a given temperature causes a loss of capacity and increase in internal resistance that is proportional to the square root of the time.  Higher SOC imparts greater degradation.  Higher temperature imparts greater degradation.  Keeping your computer plugged in all the time with the battery ensures that the battery stays hot and charged as much as is possible.  If nothing else, turn down the performance of the laptop when you don't need it to overperform so that it runs cooler.

Cycle life - while you might think that it makes sense to fully discharge the batteries, rather than put lots of little cycles into it, the opposite is true.  Lots of little cycles do virtually nothing to the cells.  100% discharges are the bane of their existance. 

So, the reality is, you could optimize battery life by keeping the battery about 50% charged, letting it discharge a bit and then recharging it, and if chargers allowed for it, floating them at a SOC that was not 100%.  However, it is much much easier to just get on with life and use your laptop and not worry too much about this.  Assume your battery costs you $1 a day.  Put that much away for its replacement.  When it is fine on the day you buy your next computer, you have now paid for that new computer, or a good size chunk of it.
A good laptop (or nearly any laptop) is not going to charge your battery constantly. Once the battery is charged, the computer will only monitor its level and recharge the battery occasionally when the level drops below a certain treshold (like 90% but it differs and can be user-set for some systems).

The effect on battery life of running constantly on AC power is minimal by my experience.

Batteries seem to suffer primarily through random and erratic partial discharges/recharges as is commonly seen in business settings where the users just don't care.

Also, batteries are made to last a certain number of discharge/recharge cycles. So you don't want to run on battery more often than needed. But most people say you should discharge/recharge occasionally, but it is not really that important by my experience.

While I agree with most of what Soumen Ranjan has responded, my personal experience with running on AC all the time differs radically:
I have just tested the battery of a laptop that is over 10 years old, has been in constant use over the time, and almost always on AC power. Before today, it had not been run on battery for the last several years.
Other than that, in the few cases over its lifetime when the laptop was run on its battery, the guidelines were followed to always let it run on battery until it asks for power (at the 10% treshold), and then let it charge.
Today, the battery lasted-in common use-for about 75% of the time it would when the laptop was new. (However, I hadn't run battery learning which could possibly improve the outcome.)
It's a Fujitsu laptop with a Panasonic Li-on battery.

Your question has a logical problem. When the machine is plugged in, it is not always charging.

Batteries are "working" when those are supplying the power to the machine or when you are charging them. When "working" they are aging the most and are aging the fastest. When the machine is plugged in into power the batteries do not need to work all the time. So if you use the machine plugged in all the time, you extend the life of the batteries (quite considerably).

This of course is valid for Apple devices, those have intelligent handling and charging of the batteries. (I do not know about the other makes, they probably also work like that.)

With Apple laptops you can check the "Cycle Count". For example this MacBookPro I am using at this moment has a Cycle Count of 282, after 4 years of very heavy usage (practically running 24 hours per day, most of the time). Almost always plugged in, since I am using it as a desktop computer, mostly in closed clamshell mode, with 2 monitors, external disks, external keyboard, mouse attached.

ON a MBP you can find cycle count here:
Click on About This Mac/System Report/Power

Laptop manufacturer's are not complete dummies, they know that people plug in laptops and leave them plugged in, so they design them to cope with constantly being connected.

Basically, they charge up to 100% and then stop. When the charge falls below a predetermined level, such as 95%, it will charge up again.

Use it or lose it. Batteries last the longest with moderate use. Use it on the battery for a bit, charge it up a bit, and so on. Regular light use is important. Leaving it at 100% or 0% for long periods will shorten the life.

Batteries are designed for a certain number of cycles - 100% charges. Running it down to 40% one day, then recharging and running down to 60% another day is one cycle because 40% + 60% = 100%.

Some batteries are designed for 1,000 cycles. That's about 5 years running it down to 50% every single day. Most people don't use it that much. Cheap batteries may not be designed to last that long though.

If your laptop overheats and damages the battery then it is faulty. It should not be designed to do this. Playing games while charging will produce a lot of heat, but providing there is plenty of ventilation there shouldn't be a problem. Old laptops might have ventilation slots and fans clogged up with dust though.

Unplug the mains power and use the battery once a week to keep the battery in good condition.

Most laptops today use lithium Ion (Li-ion) batteries. Overcharging Li-ion batteries is not a problem and does not affect the battery life span. These batteries can be charged 300 to 500 times, and they have an internal circuit to stop the charging process at full charge. The control system prevents overcharging, which can cause the lithium ion battery to overheat and potentially burn. This is why the Li-ion batteries are more expensive. The only way for the Li-ion battery to overcharge is if the charging system malfunctions, and then the battery will heat up while in the charger. If you don't plan to use your laptop for long period of time, you can extend the life of your lithium ion laptop battery by storing it with a 50 percent charge. A fully discharged battery left for a long period of time will lose its charging capacity. Fully charged batteries discharge when they are left unused, and will lose effectiveness. This is why it is recommended to discharge a Li-ion battery until it is almost out before shutting down. Running a Li-ion battery down completely will diminish capacity. Keep the battery in a cool place and do not store your computer for a long period of time while it is fully charged.

Older laptops use Nickel-cadmium (Ni-Cd) and Nickel Metal Hydride (Ni-MH) batteries. They require more maintenance than Lithium Ion batteries. They must be fully discharged and then fully recharged so that they don't lose battery life. These batteries can be overcharged only if they go through too many cycles of charging and discharging, which reduces their effectiveness and capacity. It is better to charge all batteries fully before use; then use on battery power until the optimum level for each battery type is reached, before recharging.

Source : howstuffworks
According to the answer on Apple StackExchange, you need not disconnect the charger when the battery is full. Apple's batteries are smart enough to manage it on their own.

I have noticed this myself, when the battery is full, the machine stops charging (the indicator on the charging port turns green)  and it looks to work on the incoming power. You can also check the status in System Information -> Power. It gives the information about the charging status with the full capacity, charge remaining in mAh and also the Cycle Count. The full charge capacity shown here, changes from time to time based on the use.

The batteries are basically designed to last x number of cycles. So, it would be better to keep the cycle counts as low as possible, to make the battery last longer. To know what exactly One Cycle means, read up Apple - Batteries - Why Lithium-ion? which beautifully explains it.

But at the same time, people say that it is not recommended to be used plugged in for a long period. Many people have quoted the following line from one of the old Apple's Battery page, which is now missing:
"For proper maintenance of a lithium-based battery, it's important to keep the electrons in it moving occasionally. Be sure to go through at least one charge cycle per month (charging the battery to 100% and then completely running it down)" . However, a user answers that, periodically draining the battery, only serves to allow the Battery metering systems to accurately estimate the remaining battery capacity. It is not needed to maintain the battery itself, and actually wastes a battery cycle.

So, does it mean that we can use it plugged in forever and never discharge it. Will it make the battery eternal?
NO. Because when we are not using the battery, its like storing it unused for a long time. And when keeping it unused, Apple's - Battery Maximizing Performance recommends to keep the charge in the battery around 50%, else its capacity will reduce. Look up the Table 3  on How to Prolong Lithium-based Batteries if you are interested to know why around 50%.

Also, the lithium-ion batteries are not supposed to be completely drained as many people suggest. The myths related to batteries can be read here: Debunking Battery Life Myths for Mobile Phones, Tablets, and Laptops

Some of the tips to increase the battery life are as follows:
  1. Avoid deep discharging (< 10-20%) of the battery
  2. Avoid rapid discharge of the battery i.e. on heavy usage (high graphics games), use it plugged in.

Hence, the optimum battery maintenance is  basically a balance between the number of cycles used and the capacity depreciation when its not being used!

So, its pretty difficult to maintain the performance of the battery in a long run in the normal day to day use. The batteries are anyways, not designed to last forever and dies out eventually.

tl:dr : Yes its OK, to use Macbook Air, plugged in even after the battery is full. No, It does not affect the battery in long run, provided you discharge(use) it once in awhile (which almost everyone does)

Thanks for the A2A Parag Parate. It made me ponder deep into this question which was puzzling me too :P

P.S . If you are a Geek,  you may dive into the technical details here: Lithium-based Batteries, Charging Lithium-Ion Batteries, How to Prolong Lithium-based Batteries
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