Bilingual people seem to have different neural pathways for their two languages, and both are active when either language is used. As a result, bilinguals are continuously suppressing one of their languages -- subconsciously -- in order to focus and process the relevant one.
The first evidence for this came out of an experiment in 1999, in which English--Russian bilinguals were asked to manipulate objects on a table. In Russian, they were told to "put the stamp below the cross". But the Russian word for stamp is "marka", which sounds similar to "marker", and eye-tracking revealed that the bilinguals looked back and forth between the marker pen and the stamp on the table before selecting the stamp.
And it seems the different neural patterns of a language are imprinted in our brains for ever, even if we don't speak it after we've learned it. Scans of Canadian children who had been adopted from China as preverbal babies showed neural recognition of Chinese vowels years later, even though they didn't speak a word of Chinese.