What signal processors comprise an Event-Related Potential system for EEG?
Event-related potentials are an issue in many aspects of physiology, not just in EEG analysis, so this answer is more general.

The main problem is that electrical or other signals that are associated with some "event" are typically much lower in magnitude than the background noise in the system being examined. Electrocardiograms are the exception in this regard. The trick then is to get the event-related signal out of the noise.

This is done by multiple repetitions of the "event" combined with signal averaging. You mark the electrical record with the time of each event, line up portions of the record by the event markers, and average the lined-up signals around the multiple events. This can be the full waveform for sensory evoked potentials, or if you are examining action potentials in a nerve, the histogram of post-event occurrence times of action potentials.

The particular signal-processing mechanisms used have evolved with technology. I'm old enough to have known people who 60 years ago recorded electric neural activity on magnetic tape and processed the data, after analog-to-digital conversion, on what were then state-of-the art computers with 65,000 words of memory. Even then, there was discussion about the best technologies to use for specific purposes, as this 1959 MIT monograph indicates.

The main problem, the low level of event-related signals to the noise from other electrical activity, still remains 60 years later. The noise is biologic rather than technical, so more recent improvements have more to do with ease and speed of processing rather than the fundamental signal-to-noise problem. For event-related potentials, use whatever technology allows you to collect the electrical data along with linked notations of the event times, then to average the signals synchronized on the events to build up the signal out of the noise. In a standard clinical application of brainstem auditory evoked potentials to evaluate hearing, you may need to average over 500 or more stimulus presentations.

This signal-in-noise problem is also seen in functional magnetic resonance imaging (fMRI) of brain function, which must consider not only the signal over time but also differences among brain regions within individuals and differences among individuals. The Wikipedia page on fMRI goes into some detail on methods for getting specific signals out of the background noise. You should pay particular attention to the issue of "Block versus event-related design" on that page. My understanding is that for "events" of the type that you consider, like "thinking about a turtle," the block design works best in fMRI. The paper by Thierry et al, provided in the helpful answer by @Christiaan, shows how to apply this type of blocking approach to EEG measurements during presentation of visual stimuli.

The methods in that paper by Thierry et al also indicate the type of practical care that is used in professional analyses of this sort. The authors go into detail about the nature and placement of the electrodes, choice of which electrodes to use for analysis, signal-sampling rates, and so on. For example:


  Scalp activity was digitized at a 1-kHz sampling rate from 64 Ag/AgCl electrodes distributed throughout the scalp according to the 10–20 convention using Cz as a reference. Impedances were kept below 7 kOhms. The electroencephalogram was filtered on-line between 0.01 and 200 Hz and off-line low pass at 35 Hz using a zero phase-shift digital filter. Eye blink artifacts were mathematically corrected, and signals exceeding $\pm$ 75 $\mu$V in any given epoch were automatically discarded. 


Reading papers like this are probably the best way to learn about what's involved and to find people who can teach you how to do it.