This is a question I often had in the past, I developed several tools on Excel and on this website. There are different techniques to calculate the size of your position.
note: In this page the risk will always be in percent. If we want the risk in dollar (or other currency) we'll have to make (risk x balance)

The simple way: using the maximum risk you're willing to take.

My first approach was to make it simple, this is the most common way to calculate the size of the position, you'll find the formula everywhere on the Internet. Let's say, I have $10,000 on my account, I don't want to lose more than 1%, then the size of the position is adjusted to limit the loss to $100.

For a forex account:
Position Size = (Account Balance x risk / stop Loss in pips) x (lot size / pip value)

Example: Your account balance is $10,000, the risk is 1%. You trade EURUSD (standard lot is 100,000 and the pip value $10) with a stop loss at 50 pips. The position size will be:
(10,000 * 0.01 / 50) x (100,000 / 10) = 2,000 units or 0.02 lots.

For a stock account:
Position size = Account Balance x risk / stop loss in $

Example: Your account balance is $10,000, the risk is 1%. You buy AAPL with a stop loss 2.3$ below your entry price. The position size will be:
10,000 * 0.01 / 2.3 = 43.5 stocks. you'll have to round to 43 to stay below a 1% risk.

A more evolved way: using the win rate and reward risk ratios

The previous calculation let you decide what level of risk you want to take. But you shouldn't choose an arbitrary number. The maximum risk could be defined by your historical win rate ratio and the risk reward of the trade. You won't risk the same amount if your win rate ratio is 90% (you win 9 trades out of 10) or 10%. In the same way you shouldn't risk the same amount if your reward risk ratio is 3:1 or 0.5:1

More information on the win rate ratio

First we need to evaluate what is your trade expectancy. The trade expectancy is the average amount you will win or lose per trade. It is defined by the formula:

Expectancy = (probability to win x average win amount) - (probability of loss x average loss amount)

We'll suppose that the reward risk and the risk are constant

The probability to win is the win rate ratio
The average win amount is (risk x balance) x (reward risk)
The probability of loss is (1 - win rate ratio)
The average loss amount is (risk x balance)

Expectancy = (WinRate x Risk x Balance x RewardRisk) - ((1 - WinRate) x Risk x Balance)

I suppose you want an expectancy > 0. That mean that if your WinRate is lower than 1 / (1 + RewardRisk), then you'll need to work your trading system.

Let's go deeper in the rabbit hole, if you have WinRate = 1 / (1 + RewardRisk), based on what we wrote before then your expectancy is 0. Which should mean that after a high number of trades your balance should stay at a constant level. That would be too easy.
With a simple example you'll see that not the case. We still have our $10,000 account, our Win rate ratio is 50% and the reward risk ratio is 1:1, that's not too bad, based on the formula our expectancy is 0, not perfect but we shouldn't lose any money on the long term, that's a good start. I make 100 trades, win 50 and lost 50 of them. When I win my balance increases by (1+risk) and when I lose it decreases by (1-risk). At the end I have on my account: balance x (1+risk)^50 x (1+risk)^50

I risk 1% on each trade. My new balance is = 10,000 x (1+0.01)⁵⁰ x (1-0.01)⁵⁰ = $9,950
I risk 20% on each trade. My new balance is = 10,000 x (1+0.2)⁵⁰ x (1-0.2)⁵⁰ = $1,299

As you can see, it's not that simple. When you lose 1% you'll have to make more than 1% to regain your lost. A simpler example, if you lose 50% of your account you'll have to earn +100% to regain your lost: you have $100 if you lose 50% then you have $50. At this point you have to earn +100% to come back to $100.

The formula behind those numbers is:

(1 - Risk)^(1/WinRate -1) x (1+ Risk x RewardRisk) = 1

Than mean:

RewardRisk = f(Risk,WinRate) = ((1 - Risk)^(1 - 1/WinRate) - 1) / Risk
WinRate = f(Risk,RewardRisk) = 1 / (log[1-R](1 / (1 + Risk x RewardRisk))) + 1

Unfortunately we can't have the Risk as a function of RewardRisk and WinRate. But if you have the values of WinRate and RewardRisk it's easy to estimate the Risk. whew!

I told you, that's not as easy as it seems at the beginning, but don't worry, we have forms that'll do the job for you.

A better way: using the win rate and reward risk ratios and R-Multiple

Stay here, it's not finished. As you probably know, the theory is not the practice and vice versa. In the real life we rarely open a trade and let it alone, we can move the stop loss or take profit, we can manually close the position, we also may take a partial profit. The reward defined by the RewardRisk we have at the opening of the trade may be different that the Reward we have when the trade is closed. That's where the R-Multiple appears. R in R-Multiple is the initial risk, for example if you're fixed your stop loss to limit the risk at 1% then R=1%, and the R-Multiple is the exit (win or loss) calculated in terms of R. In this case if you exit with a 1.5% then R-Multiple = 1.5.
That's a similar concept as RewardRisk ratio. But RewardRisk is theorical and R-Multiple is what happened on your account.

On your portfolio you'll have to take into consideration those two data. If your R-Multiple is always equal to your RewardRisk ratio, then no problem. But if it's constantly lower then you'll have to include this information when you calculate the risk you take on your trades.

An example: your RewardRisk Ratio is 2:1, but you take a profit on half of your position when the price hit 50% of your target point. You buy at 1.0000 with a stop at 0.9000 and a target at 1.2000, you take a profit at 1.1000 and the rest at 1.2000. The risk was 0.1000, You win 50% x 0.1000 + 50% x 0.2000 = 0.15. R-Multiple = 1.5:1. To be sure that your Risk is well adjusted when you calculate the size of your position you shouldn't take 2:1 as a RewardRisk ratio but 1.5:1.

Here again, we have some tools that will calculate your RewardRisk and R-Multiple based on your trades.

The ultimate way: using all the data above depending on market conditions

Take a deep breath, we are almost at the end. All of this would be really useful if each of your trades had the same R-Multiple and the WinRate was constant. But... It's probably not the case. And this is where we need to go deeper in the analysis of our portfolio and get the WinRate based on the market conditions. You probably don't have the same value when the market is flat or trending, when the volatility is high or low... You probably don't have the same success on GBPUSD and EURJPY.
That what we may call data science, all of our research are focused on this matter. We are working to provide a trading platform that will bring you all the useful information when you need it.