Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. Something left over after other parts have been taken away.
 n. Mathematics The number left over when one integer is divided by another: The remainder plus the product of the quotient times the divisor equals the dividend.
 n. Mathematics The number obtained when one number is subtracted from another; the difference.
 n. Law An estate in land that is conveyed only after the termination of a preceding estate created at the same time.
 n. A book that remains with a publisher after sales have fallen off, usually sold at a reduced price.
 transitive v. To sell or dispose of as a remainder.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A part or parts remaining after some has/have been removed.
 n. The amount left over after subtracting the divisor as many times as possible from the dividend without producing a negative result. If n. (dividend) and d (divisor) are integers, then n. can always be expressed in the form n = dq + r, where q (quotient) and r (remainder) are also integers and 0 ≤ r < d.
 n. The number left over after a simple subtraction
 n. Excessive stock items left unsold and subject to reduction in price.
 n. An estate in expectancy which only comes in its heir's possession after an estate created by the same instrument has been determined
 adj. remaining
 v. To mark or declare items left unsold as subject to reduction in price.
from the GNU version of the Collaborative International Dictionary of English
 adj. Remaining; left; left over; refuse.
 n. Anything that remains, or is left, after the separation and removal of a part; residue; remnant.
 n. The quantity or sum that is left after subtraction, or after any deduction.
 n. An estate in expectancy, generally in land, which becomes an estate in possession upon the determination of a particular prior estate, created at the same time, and by the same instrument; for example, if land be conveyed to A for life, and on his death to B, A's life interest is a particuar estate, and B's interest is a remainder, or estate in remainder.
from The Century Dictionary and Cyclopedia
 n. That which remains; anything left after the separation, removal, destruction, or passing of a part.
 n. In mathematics, the sum or quantity left after subtraction or after any deduction; also, the part remaining over after division: thus, if 19 be divided by 4, the remainder is 3, because 19 is three more than an exact multiple of 4. In the old arithmetics called the remainer.
 n. In law, a future estate so created as to take effect in possession and enjoyment after another estate (as a lifeinterest) is determined; a remnant of an estate in land, depending upon a particular prior estate, created at the same time, and by the same instrument, and limited to arise immediately on the determination of that estate, (Kent.)
 n. In the publishing trade, that which remains of an edition the sale of which has practically ceased, and which is sold out at a reduced price.
 n. Synonyms Rest, Remainder, Remnant, Residue, Balance, Rest is the most general term; it may represent a large or a small part. Remainder and residue generally represent a comparatively small part, and remnant a part not only very small, but of little or no account. Rest may be applied to persons as freely as to things; remainder and residue only to things; but we may speak of the remainder of a party. Remnant and residue are favorite words in the Bible for rest or remainder, as in Mat. xxii. 6 and Isa. xxi. 17, but such use of them in application to persons is now antique. Balance cannot, literally or by legitimate figure, be used for rest or remainder: we say the balance of the time, week, space, party, money. It is a cant word of trade.
 Remaining; refuse; left.
 n. The right to succeed to a title or position on the decease of the holder; especially the right of succession to a peerage expressly assigned to a certain person or line of descent in default of male issue in the direct line.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. the number that remains after subtraction; the number that when added to the subtrahend gives the minuend
 n. the part of the dividend that is left over when the dividend is not evenly divisible by the divisor
 v. sell cheaply as remainders
 n. something left after other parts have been taken away
 n. a piece of cloth that is left over after the rest has been used or sold
Etymologies
Examples

Losing, say, 6.5% of your contributions outright but earning 7% on the remainder is the same as earning 0% on the whole lot.
Comment of the Week, Arnold Kling  EconLog  Library of Economics and Liberty

You will see that the first 300 kilowatt hours or so are charged at more economical rates, closer to 5 US cents per KW hour, but that the remainder is at this very high rate.

The remainder is taken by the university to cover operating budgets and administrative costs.

The remainder is paid from the general fund, in longstanding recognition of the value to the general public of having a safe, efficient air transportation system (which, BTW, is the envy of the rest of the world, including Canada).

Roby got about 70% (appx. $253,000) of her contributions from individual donors, the remainder from the party or PACs

It offers two electricity tariffs – one from 100% renewable sources, and one where 41% of the power comes from its wind turbines, and the remainder is conventionally sourced power.

Mix a quarter of whipped cream into batter, then fold in remainder just until combined.

“The remainder is linked to the products we sell, so we've been working for some time with suppliers and challenging them to make products and packaging with the environment in mind.”
WalMart Canada to Eliminate PVC Plastic in CFL Light Bulb Packaging

Lowland swamp forests cover approximately half of the park whilst the remainder is lowland through to montane tropical rainforest.

Its not good enough any more to tell consumers 25% of the products are made in China and other countries and the remainder is made in Canada.
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