Is it possible to process electrical signals from the brain and interpret the results as exact thoughts?
The brain activity is electric and chemical. The male adult human brain contains about 86 billion neurons (Azevedo et al). There is about 100 trillion connections between them. Solving a puzzle like that is not easy..


  what sort of breakthroughs would be necessary to intercept these signals and interpret them as exact thoughts?


What you are referring to would be called solving the neural code in neuroscience. With today's methods, it is probably not possible.

Consider electroencephalography (EEG). It records the electrical activity of the brain. Ag/AgCl sensors are placed along the scalp (typically 64-256 in research settings). When about 50,000 parallel neurons fire simultaneously, a change in a recorded signal can be seen. While the time resolution is about 1 ms for EEG, the spatial resolution is several centimeters (it is not easy to find which areas produced the recorded signals; it is an inverse problem). A newer technique, called magnetoencephalography (MEG), is becoming more and more used but is expensive. It records the magnetic fields related to the electrical activity and allows better localization.

Is there other complications? Well... lot's of them. Blink your eyes during the recording and there will be a major artefact masking the brain signals in most channels (10-100x greater in amplitude than the brain signals).

Anyway, you would probably be interested in the new research involving the use of machine learning techniques: it has been possible to decode the contents of dreams, reconstruct what a subject is seeing, etc. The best papers have been published by Science, Nature, and PNAS, so using their search engines with the keywords decoding and brain should allow you to explore the topic easily.

PS. I did not have time discuss fMRI; someone else can may be do that..