There are a variety of Fourier Transforms (and Series) such as Continuous Time FT, Discrete Time FT, Discrete FT all of which are generally attributed to the fundamental assertion made by J.B.Fourier about at the beginning of 19th century, which claims (without proof) that "if a continuous-time signal (function) x(t) is periodic with T, then it is possible to represent that signal x(t) as an infinite sum of harmonically related trigonometric functions (sines and cosines) as $ x(t)= \sum{ a_k \sin { 2\pi kt \over T} + b_k \cos { 2\pi kt \over T}}$ in which the weights $a_k$ and $b_k$ (the Fourier Coefficients) represent the amount of that particular harmonic in the signal x(t) being analysed"
In fact the above argumentation is strictly for Continuous-Time Fourier Series. But the core idea is generalised into Fourier-Transforms of aperiodic and periodic (with the help of impulse $\delta (t)$ functions) signals. There are conditions on which signals can have such a representation.
In essence, computing a Fourier Transform means finding those coefficients $a_k$ and $b_k$ for which the method is suggested by the analysis equation of the Fourier transform, while the equality in the first paragraph is noted as the synthesis equation.
Eventhough it is quite solid to understand the meaning of those sinusoids inside a periodic signal, when it comes to non-periodic signals, for which we use the Fourier Transforms, the exact meaning of what a single sine wave represents inside such a signal is a little vague and instead we emphasize the transient character of that signal under concern and the necessity of existance of a continuum of infinetely many sine waves.
