In my stock account (TDAmeritrade if it matters), there are many transactions with description "MONEY MARKET PURCHASE", "MONEY MARKET REDEMPTION", "MONEY MARKET REDEMPTION (MMDA1)". What are they? Also, strangely some of them have amount as '0.00' or '0.01'. Why the (nearly) nil amount?
A money market fund is a type of sweep account.
Brokerages use sweep accounts to take advantage of the cash you let sit there, so they(the brokerage) can make money on the interest. This is done automatically(typically daily) so whatever percentage or thresholds they want to use is automatically applied after you've added or subtracted cash into your account.
In the case with the zero or near-zero amounts being shown, I assume this is for transparency and the result of the automated process to remove the money to their "money market fund" where they have control of it i.e. earn interest on the cash that otherwise be idle. If you were to pull the cash out yourself though you'd have no problem doing so, they'd automatically transfer the correct amount, you just get to see that happening with
"MONEY MARKET PURCHASE", "MONEY MARKET REDEMPTION", "MONEY MARKET REDEMPTION (MMDA1)"
unlike your checking account, though the same concept is being used: stored money is invested to make money on that money at no risk (well FDIC limitations) to the person storing it.
A sweep account automatically transfers cash funds into a safe but higher interest-earning investment option at the close of each business day, e.g. into a money market fund.
Sweep accounts try to minimize idle cash drag by capitalizing on the immediate availability of high-interest accounts.