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My hope from this question is to become a repository of the behavioural characteristics, use-cases and interesting features of the key Ito processes used in quantitative finance - examples being GBM, OU, CIR etc.
I don't think there is a need to derive the processes from their SDEs; the focus is more on the peculiar characteristics and behaviours that make them useful under different scenarios.
I will provide some references such that you can see where the different processes are used. These papers typically motivate their models and show which effect the single paramaters have and what asset price dynamics the model intends to capture.
The geometric Brownian motion is the easiest model for exponential growth with stochastic noise. The randomness is driven by a single Brownian motion. Here you always get continuous sample paths which are positive. The process is furthermore a Markov process (i.e. memoryless) which kind of agrees with market efficiency. At each time point, the geometric Brownian is log-normally distributed. The GBM is of course the key assumption of the Black-Scholes model. However, the geometric Brownian motion performs empirically rather poor. One of the earlier papers on stochastic volatility, Hull and White (1987) uses the geometric Brownian motion as model for the variance. Dothan (1978) models the short rate with this process.
It is possible to add jumps to a geometric Brownian motion yielding a stochastic process with discontinuous sample paths. With such a process, you get single, large jumps which occur occasionally. Merton assumes jumps arrive with a Poisson distribution whereas the jump size is normally distributed. Kou (2002) models the jump size with a double exponential distribution which gives an up jump a different mean than a down jump. In general, the jumps increase the likelihood of ``extreme events'' and hence lead to fatter tails which improves the empirical performance.
These processes are not driven by a Brownian motion and are completely discontinuous, i.e. they only move via jumps. Their density functions are not always known is closed-form but their characteristic function is typically quite easy such that Fourier methods are straightforward to apply. An easy way of constructing these processes is through subordination, i.e. you change the calender time to business time. These processes naturally can give you a skewed distribution with heavy tails which fits empirical data quite well. Examples include the Variance Gamma Process from Madan and Seneta (1990) and Madan, Carr and Chang (1998), the normal inverse Gaussian process from BarndorffNielsen (1997), the CGMY model from Carr, Geman, Madan and Yor (2002) and finite moment log stable model from Carr and Wu (2003).
This process is mean-reverting and hence used to model volatility and interest rates. Vasicek (1976) uses the OU process for short rates. Ho and Lee (1986) and Hull White (1990) generalise this model. The OU process is also used by Stein and Stein (1991) and Schöbel and Zhu (1999) for the volatility. The exponential Vasicek model assumes that the log of the short rate follows an Ornstein Uhlenbeck process. The OU process is at each time point normally distributed and hence may be negative with positive probability.
The CIR process is also mean-reverting and used by Cox Ingersoll Ross (1985) for the short rate. Heston (1993) uses the process to model the variance. Christoffersen, Heston and Jacobs (2009) model the variance as the sum of two uncorrelated CIR processes (double Heston model). The CIR process follows a chi squared distribution and thus, it is always non-negative. You can think of this as follows: tha variance of aCIR process $(X_t)$ is proportional to $\sqrt{X_t}$ and hence, if $X_t$ comes close to zero, so does its variance. Then, the mean-reversion pulls $X_t$ away from zero.
One of the earliest model from Bachelier (1900) uses an arithmetic Brownian motion in order to price rentes. Cox, Ross and Rubinstein (1979) use a discrete-time process to build their tree. The CEV from Cox (1975) is used for stocks and SABR model from Hagan, Kumar, Lesniewski and Woodward (2002) may be used for interest rate models. Stochastic volatility models similar to the CIR process are the 3/2 model from Heston (1997) and the 4/2 model from Grasselli (2017). Bates (1996) combines the Heston and Merton model to get an SVJ model. One can combine the pure jump process with stochastic volatility and obtain stochastic clock models. A special case is the mixture of a variance gamma process with the CIR process which gives the VGSA model from Carr, Geman, Madan and Yor (2003). In general, considering several processes at the same time gives you the possibility to play with correlations which again affects skewness and kurtosis. Local volatility models use the SDE of a geometric Brownian motion but consider the volatility as a deterministic function of time and stock price. This local volatility function is found by calibration, see Derman and Kani (1994) and Dupire (1994).
So a lot of time and work has been spent on proposing various models. The question is always what price do you want to model and what characteristics of it are important for your applications. You have a trade-off between the model being simple enough to price complicated products and to capture all important dynamics of the underlying asset.