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So I always understood the error term to be the difference between the observed value from the true, yet unobservable, function value. However, I often here, especially related to the economics fields, articles and people talking about error terms as 'shocks'.
Can anyone offer an intuitive, non-math intensive, explanation?
We are talking about transitory (temporary) shocks. Permanent Shocks create structural shifts, and should not be captured by the error term.
For temporary shocks, take for example a toy-specificaton for demand for money (i.e for liquidity), under controlled interest rates and flexible money supply:
$$M^d = AY^ke^{-a\mathbf i}$$
where $Y$ is output and $\mathbf i$ is the nominal interest rate. In a linearized econometric setting we would get
$$\ln M^d = \ln A +k\ln Y -a\mathbf i +u$$
where $u$ is the error term -excuse me, the shock. What shock?
Our specification by design does not take into account fluctuations in precautionary demand for money. This comes about due to uncertainty / expectations for future events: if, say, news arrive that a banking sector crisis is imminent, people will tend to get their savings out of the banks, thus increasing abruptly the demand for money (interest bearing accounts are not money, they are assets). This abrupt fluctuation will be included in the "error term".
ADDENDUM: Permanent Shocks
In response to an OP' comment, assume now that a change in the technology of transactions happens. For example, the telephone (I mean the "traditional" telephone, not cell phones). This implies that now you can call a shop, find out if it has the product you need, order one and arrange for a specific date when you will appear in the shop, give the money, take the product. Previously, you had to have the money on you and to go physically in the market place to find what you wanted -and sometimes you wouldn't find it, and you would have to go again the next day: this means that the amount of money for the transaction would remain money and not being re-transformed into an interest-bearing asset, for a longer time period.
Overall, this reduces the demand for money, for given output and interest rates: it permanently reduces the constant term $A$. In that sense, it is a structural shift, and should not be left to be conceptually captured by the error term, which reflects something that affects the dependent variable only in a certain period. Even if the error term is autocorrelated (=> current shocks propagate into the future), still this propagation will eventually die out, and so strictly speaking, again, it is not the right way to deal with structural shifts
