There are two parts to your question which I try to answer separately. The first one is about what calibration actually is whereas the second question deals with risk-neutral pricing.
As an example, we can use any model. I continuously refer to the stochastic volatility model from Heston (1993) as an example for equity options. Any thoughts equally apply to other models or asset classes (think of interest rate derivatives and short rate models). The Heston model for a stock price $S_t$ reads as
\begin{align*}
\text{d}S_t&=\mu S_t\text{d}t+\sqrt{v_t}S_t\text{d}W_S, \\
\text{d}v_t&= \kappa(\theta-v_t)\text{d}t+\xi\sqrt{v_t}\text{d}W_v,
\end{align*}
where $\text{d}W_S\text{d}W_v=\rho\text{d}t$. Thus, there are five model parameters ($\mu,\kappa,\theta,\xi,\rho$).
Estimation and Calibration
Estimation finds the real-world distribution of stock returns; calibration aims to parametrise the risk-neutral distribution. To this end, estimation uses historical observations whereas calibration uses observed market prices. Estimation is commonly used for risk management (e.g., value-at-risk) whereas calibration is used for pricing/hedging/trading derivatives.
Let's begin with estimation: Take historical returns of the S&P 500 index and compute the (log-)likelihood function from the Heston model to find the parameters that are most likely to have given rise to the given sample of stock returns (This is called maximum likelihood estimation). A generalisation is GMM which compares sample moments from the data with moments implied by the model and finds the parameters that minimise the difference between the two. Intuitively, you seek parameters such that simulations of the model yield sample paths that look very similar to the actual data you observe in financial markets. In particular, this includes that asset prices grow at same rate $\mu$ that reflects the systematic risk of that asset. You could such a model for forecasting.
Let's turn to calibration: Take current market prices for S&P vanilla call and put options. Using the closed-form solution for option prices in the Heston model, you now seek model parameters which minimise the (squared) difference between observed market prices and implied model prices. This time, you don't care about historical return data. You only take yesterday's option prices and you match your model with these prices. If you calibrate a short rate model, you can use the prices of zero bonds, bond options, caps etc. Your choice of calibration instruments should match your intended application of the model. Because these instruments are priced using a risk-neutral framework (see below), the calibrated parameters correspond to the $\mathbb{Q}$ measure in which assets are assumed to grow at rate $r$ (and not $\mu$). Thus, these parameters must not be used for forecasting! These parameters, instead, are used for valuing other (complex) derivatives.
Two points are in order
There are two different time horizons: Estimation requires a (long) time series of returns. Calibration requires price data from one single day.
In practice, there are many subtitles when one wants to implement estimation or calibration in practice: How to discretise a continuous time model? What moments to match for GMM? How to weight different market prices? How to clean options data? Calibrate to prices or implied volatilities, etc.
Pricing and Risk-Neutral Distribution
Option pricing is (almost) always done under the risk-neutral ($\mathbb{Q}$) measure. The value of a European-style call option is computed as discounted expected payoff
\begin{align*}
C=e^{-rT}\mathbb{E}^\mathbb{Q}[\max\{S_T-K,0\}].
\end{align*}
The risk-neutral distribution ($\mathbb{Q}$) is necessary such that we can discount at the risk-free rate $r$ (instead of bothering with risk premiums embedded $\mu$). The idea is that the $\mathbb{P}$ distribution (i.e., real stock price movements) contains these risk premiums which are extremely difficult to identify. The risk-neutral pricing framework allows us to avoid all of this and arrive at the same option prices while pretending everyone is risk-neutral (we pretend that $\mu=r$).
The crux is that we do not need to know $\mu$ to price options. When we calibrate a model, we minimise the difference between market prices and model prices. These model prices are computed using the risk-neutral framework (all option prices are). Thus, you only recover the risk-neutral distribution from option prices, see also Breeden and Litzenberger (1978). But that's no problem. We only need that risk-neutral distribution to value derivatives.
The $\mathbb{P}$ and $\mathbb{Q}$ distribution are linked via the stochastic discount factor (or pricing kernel) which is just a scaled Radon-Nikodym derivative (change of measure via Girsanov's theorem). Thus, the $\mathbb{P}$ parameters and the $\mathbb{Q}$ parameters are also linked, using the risk premiums embedded in the pricing kernel, see this answer. Thus, if we knew the true SDF and the true $\mathbb{Q}$ parameters, we could recover the true $\mathbb{P}$ parameters.
