As Christian pointed out, meditation means a lot of different things. Even varieties of meditation differ within themselves; according to one account, mindfulness meditation practices differ along 5 dimensions:
(1) intention and context of mindfulness practice, (2) bare attention, (3) attentional control, (4) wholesome emotions, and (5) ethical discernment
This leads to different effects depending on the style of meditation. Nonetheless, here are some of the findings:
Meditation has been shown to be effective in self-regulation of many psychology factors including
present-moment pain, negative body image, inhibition of activity by pain, symptoms, mood disturbance, and psychological symptomatology, including anxiety and depression.
Transcendental Meditation has been shown to be particularly effective in improving anxiety, though other forms of meditation also helped, to a lesser degree.
Another review concluded that:
Mindfulness-based therapeutic interventions appear to be effective in the treatment of depression, anxiety, psychosis, borderline personality disorder and suicidal/self-harm behaviour.
Also, ACT therapy, a relatively new form of cognitive-behavioural therapy based on mindfulness, has been shown to be effective in the treatment of a number of issues, including psychosis, anxiety, depression, pain management, quitting smoking, and substance abuse. A re-analysis showed that ACT was even more effective than some existing treatments.
In contrast, another meta-analysis concluded that the effect of the mindfulness-based stress reduction program on depression and anxiety is not reliable:
Evidence for a beneficial effect of MBSR on depression and anxiety was equivocal. When active control groups were used, MBSR did not show an effect on depression and anxiety. Adherence to the MBSR program was infrequently assessed. Where it was assessed, the relation between practising mindfulness and changes in depression and anxiety was equivocal.
This massive review summarises the research on neuroimaging studies of meditation and discusses the findings as they relate to psychological and clinical effects. The studies done to date show a number of changes to different areas of the brain, including frontal and prefrontal areas, which is linked with improvements to attention and information processing speed and efficiency. They conclude that effects on brain and nervous system function are undeniable, but due to differences in practice and research the exact nature of those effects requires more study.
This review summarises some of the physiological effects linked directly to meditation (as opposed to those caused by e.g., simply resting), including changes to heart rate, blood flow, and metabolic processes.