How DRAM refresh cycles work?
Simplifying a little bit, think of DRAM as being a 2D array of memory cells1. Each cell in the array is a minuscule capacitor.

Along one edge of that array, we have a set of sense amps. There's one sense amp for each cell along that dimension. For either a refresh cycle or a normal read, we activate an entire row (or column, if you prefer to look at it the other direction). When we do that, we read the charge from that row (column) of capacitors into the sense amps. In doing that, we've drained (at least most of) the charge out of the capacitors that make up the memory cells themselves.

That will typically give somewhere in the range of a few hundred to a few thousand bits of data that are sitting in the sense amps. We can then read some of that data out of the sense amps to do to the outside world (if this was a read cycle) or we can just write it back to the memory cells (if it was a write cycle). Or, in the case of a refresh cycle, we read the data from the cells into the sense amps, then turn around and write it back out from the sense amps to the cells.

If the array were square, this would mean the amount of auxiliary storage needed (i.e., the number of sense amps) would be approximately the square root of the total number of bits. In reality, however, the array doesn't have to be square--we can choose a number that's convenient, and set the size in the other dimension to give the total amount of storage desired.

The number of sense amps does have some ramifications with regard to speed though. In particular, it's relatively slow to read data from the capacitors into the sense amps, and much faster to read from the sense amps to the outside world. This is what leads to the typical speed profile for DRAM sticks and such. Most will have a burst length of (say) 8, with a transfer profile that looks something like 17-1-1-1-1-1-1-1. That is, the first word will be transferred 17 cycles after the DRAM receives the (last element of) the address, then another word will be transferred each subsequent clock until the entire burst of 8 is completed.

The number in that initial position varies widely depending on the vintage of DRAM you're talking about. The reality is that the transfer from the capacitors to the sense amps has remained fairly close to constant for quite a long time. As clock speeds have risen, the number of clocks has risen with it to allow for the total time necessary for that transfer.

So, that first (long) transfer time is basically telling us the amount of time needed to transfer the data from the capacitor array into the sense amps. The subsequent transfers are reading data from the sense amps and sending them to the external bus. Of course, in the case of a refresh cycle the data is never written to the external bus. It's just read from the capacitors into the sense amps, then written back from the sense amps into the capacitors2. 

That being the case, in a typical commodity DRAM, maximizing speed means ensuring that there are at least as many sense amps as there are bits in a burst. If we're dealing with 64-bit words and 8-word bursts, we want (at least) 64x8 = 512 bits of sense amps. Having more sense amps than that isn't necessarily going to gain a whole lot of speed.

So that argues in favor of the DRAM array being arranged as an Nx512 bit array, with the sense amps along the 512-bit side. A read (or refresh) then consists of activating one of those 512-bit rows/columns in the array, transferring that data to the sense amps, transferring the result out the external bus (if it's a read) and writing the data back to the capacitors.

For current large memory systems, things are a little more complex than that. Rather than a single array of N x 512 bits, the memory is further broken up into banks. For example, a 1 gigabyte DRAM might consist of, say, 8 banks of 1/8th gigabyte apiece (offhand, I don't remember the standard size for a DDR3 or DDR4 bank, but that's at least in the general ballpark).




In reality, in a modern system, there are typically (what could at least be thought of as) more dimensions than that, but 2D is enough to explain the basic idea.
Sense amps are analog amplifiers. They're typically differential amplifiers. In a typical case, the DRAM will contain some dummy cells. To start a read cycle, you pre-charge those dummy cells to (approximately) half the voltage initially stored in a normal cell (or sometimes, to the full voltage, but the dummy cells have half the capacitance). You then read in the charge from the dummy cell and the normal cell, and amplify the difference between the two, so if the cell contained less than the dummy cell, the result will be driven (close) to 0, and if it's greater than the dummy cell, it'll be driven (close) to Vcc. That (now digital) value is then stored into a flip-flop, latch, etc. The dummy cells are used (instead of, for example, just feeding Vcc/2 directly to one of the inputs of the sense amp) to semi-automatically compensate for things like losses in the bit lines, which can vary both across a chip and with the current operating conditions.