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In applied mathematics, the phrase analytical methods refers to using manipulation of formula and equations (using rules of algebra, trigonometry and calculus, etc.) to solve for a particular variable, giving yourself a straight-forward expression to calculate its value. This is in contrast to numerical methods, which avoid having to solve for the variable of interest through use of a lot of arithmetic. For example, when trying to find the numerical value of a definite integral, analytical methods consists of integrating (e.g., taking the antiderivative), and plugging in numbers in the resulting formula, whereas a numerical method might be dividing the range between the bounds into small integrals, using simple linear approximations for each and summing them.
While numerical methods often result in a bit of mathematical approximation, analytical methods may use mathematical approximations as well, such as series approximations. In both cases, approximations of the physics (e.g., leaving out forces of small significance) are often used to make a problem tractable. A back-of-the-envelope calculation is an extreme case of this. Often, an analytical approximation is produced, then numerical methods are used for greater accuracy.
The term analytical methods is sometimes used more specifically: for methods dependent on branches of mathematics termed analysis:
The term analytical methods in physical sciences is also used for a different concept, i.e., to refer to techniques for analysis of something physical. The phrase is definitely used in this sense within chemistry, i.e., tests that determine what something consists of. This sense is naturally used in any science that involves some chemistry, such as biology, medicine, and astrophysics.