fMRI Processing/Analysis Overview

How is fMRI data processed and analyzed?  
After acquiring fMRI and anatomic scans, several steps are needed to transform this raw data into meaningful information. In summary, the main stages are:​
  1. Preprocessing of raw fMRI data involves recognition of outlier data followed by multiple steps to correct for noise, motion, signal drifts, slice timing discrepancies, and spatial distortions. The preprocessed BOLD data can then be overlaid on the subject's own anatomic brain images (coregistration) or warped onto a generic brain template (normalization). 
  2. Statistical analysis can then be performed using a variety of methods, the most common being a single voxel-based matrix algebra approach known as the General Linear Model (GLM). Other methods (more commonly used for resting-state fMRI) include correlation analysis and independent component analysis (ICA).  Although sophisticated analyses are typically performed "off-line" after completion of data collection, basic statistical maps can be incrementally computed and displayed while fMRI data is being acquired. Such real-time analysis permits immediate identification of data collection problems, allowing scans to be repeated while the patient is still in the scanner.
  3. Maps are then created to show brain areas whose levels of activation correlate most strongly with experimental manipulations, behavior, performance, or the activity of other brain regions. Data from a single person, scanned repeatedly under different experimental conditions, may be used to created single-subject maps. For population-based studies, data from multiple individuals may be combined and displayed as group-level maps. More sophisticated overlays can be created using mesh-based surface reconstruction, tissue segmentation, and cortex inflation or flattening. Functional connectivity maps using seed-based or ICA correlation techniques may be generated, sometimes displayed as graphs with nodes and links. 
  4. Interpretation of results depends on the experimental design and must always be placed in the context of established neuroanatomy and neurophysiology. In forward inference experiments, we attempt to conclude that a certain task (e.g. finger tapping) produces a specific region of BOLD activation. In reverse inference experiments the converse is attempted — to conclude that BOLD activation of a certain brain region means the subject is doing or feeling something (e.g. anger, pain). Reverse inference experiments are much trickier to perform and interpret; the literature is replete with false conclusions drawn from them. The third class of experiments, multivoxel pattern analysis (MVPA), uses a combination of forward and reverse inference. Machine learning may be employed first to classify activation patterns from various stimuli, then later used to predict stimuli from observed patterns of brain activity. 
These steps will all be described in much more detail in subsequent Q&A's.

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References
     Chen JE, Glover GH. Functional magnetic imaging methods. Neuropsychol Rev 2015; 25:289-313.​
     Kosslyn SM. If neuroimaging is the answer, what is the question? Phil Trans R Soc Lond B 1999; 354:1283-1294. (A good philosophical discussion on the types of information that can, and cannot be derived from fMRI experiments).
     ​Poldrack RA. Can cognitive processes be inferred from neuroimaging data? Trends Cogn Sci 2006; 10:59–63. (describes forward and reverse inference concepts)
Related Questions
     How do you design a BOLD/fMRI study?
     How are those activation "blobs" on an fMRI image created, and what exactly do they represent?