In Sensitivity Analysis the market risk is quantified by a small set of risk factors.
The instrument evaluation is then done by a Taylor expansion in the risk factors with the "Sensitivities" being the expansion coefficients. The main sensitivities are the first (Delta) and the second (Gamma) derivative of the instrument price with respect to the main risk driver. The expansion is in some cases enhanced by the first derivative with respect to the volatility of the corresponding main risk driver (Vega) or to a secondary risk driver (e.g. Rho for the interest sensitivity). Higher order derivatives like "Volga" or "Speed", cross derivatives like "Vanna", "Charm" or "Color" are rarely integrated into Sensitivity Analysis, as well as explicit time dependence (Theta), which is generally not desirable since risk controlling assumes instant shifts that neglect aging of instruments.
In many cases the relevant sensitivities can be calculated in a front-office system that holds full valuation formulas (Black-Scholes, etc.) or calculates numeric derivatives. The risk engine doesn't have to receive and evaluate reference data for a Full-Valuation Approach, but operates on the mere sensitivities. The uniformity of the evaluation function opens the additional benefit of being able to aggregate portfolio sensitivities on individual risk factors in order to increase transparency or even to trigger risk mitigation actions.
The obvious downside of Sensitivity Analysis is that higher order terms of the expansion are neglected, which may add up to notable errors, e.g. in the case of a protective-put-like payoff, where first order will systematically overestimate risk, while second order will systematically underestimate market risk.
Nonetheless, Sensitivity Analysis is a very reliable method since it is less dependent on the quality of historical data which has to be collected only for a smaller set of risk factors. Generally, it requires a smaller numerical cost than a Full-Valuation Approach.