The terminal value of an investment is the last net cash flow that is received from the investment. The contribution of the terminal value to the NPV is dependent on the value of the terminal value, the discount rate that is being used and the time period after which the investment can be considered to have come to an end.
The cases where the terminal value of the investment would be relatively unimportant is when its discounted value is very small. For example, if one buys a stock that offers a dividend of 5% and there is no expected rise in the price of the stock as the firm is one where there is no scope for expansion and all the income earned is distributed as dividend. In such a case if the stock is being kept for perpetuity, the terminal value can be ignored as it would be almost negligible.
It is not clear what relatively important exactly means and that makes it difficult to give real life examples of investments where the terminal value can be neglected. In most real life investments the terminal value would have to be considered to obtain accurate returns on the investment.