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Cosmology concerns itself with the fundamental questions about the formation, structure, and evolution of the Universe as a whole. Cosmic microwave background (CMB) radiation is one of the foremost pillars of physical cosmology. Joint analyses of CMB and other astronomical observations are able to determine with ever increasing precision the value of the fundamental cosmological parameters and to provide us with valuable insight about the dynamics of the Universe in evolution. The CMB radiation is a relic of the hot and dense first moments of the Universe: a extraordinarily homogeneous and isotropic blackbody radiation, which shows small temperature anisotropies that are the key for understanding the conditions of the primitive Universe, testing cosmological models and probing fundamental physics at the very dawn of time. CMB observations are obtained by imaging of the sky at microwave wavelengths. However, the CMB signal is mixed with other astrophysical signals of both Galactic and extragalactic origin. To properly exploit the cosmological information contained in CMB images, they must be cleansed of these other astrophysical emissions first. Blind source separation (BSS) has been a very active field in the last few years. Conversely, the term "compact sources" is often used in the CMB literature referring to spatially bounded, small features in the images, such as galaxies and galaxy clusters. Compact sources and diffuse sources are usually treated separately in CMB image processing. We devote this tutorial to the case of compact sources. Many of the compact source-detection techniques that are widespread inmost fields of astronomy are not easily applicable to CMB images. In this tutorial, we present an overview of the fundamentals of compact object detection theory keeping in mind at every moment these particularities. Throughout the article, we briefly consider Bayesian object detection, model selection, optimal linear filtering, nonlinear filtering, and multif- requency detection of compact sources in CMB images. This article's goal is to present a tutorial on the detection, parameter estimation and statistical analysis of compact sources (far galaxies, galaxy clusters, and Galactic dense emission regions) in CMB observations.