On a dial-up Internet connection, why is the speed is limited to 56 kbits/s compared to a broadband Internet connection which may carry 10 times as much as dial up through that same telephone line?
Is it because the dial up is limited to 56 kbits/s by the ISP? Does the ISP amplify the speed when you order broadband?
Dial up connections use the voice circuit for data transfer so the bandwidth is limited to bandwidth of voice channel, whereas DSL uses a separate frequency range for data which is much broader than voice bandwidth (hence the term Broadband). DSL uses a splitter to separate the voice and data frequencies and therefore both can work simultaneously.
Basically, the telephony line is limited to 64 kbits/s for each channel (8 kHz channel with PCM modulation on 8 bits per Hz, making 8 kHz x 8 bits = 64 kbit/s). Unless you use another modulation(QAM for example) or more bandwidth(more than 8 kHz telephony channel, up to MHz for example), your transmission capacity will be limited to the signal/noise ratio of your telephony channel (low values of S/N will reduce your 64 kbits/s transmission capacity). Let me explain you:
According to Shannon–Hartley theorem:
The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free digital data (that is, information) that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density.
Or: 
where
C is the channel capacity in bits per second;
B is the bandwidth of the channel in hertz (passband bandwidth in case of a modulated signal);
S is the total received signal power over the bandwidth (in case of a modulated signal, often denoted C, i.e. modulated carrier), measured in watt or volt2;
N is the total noise or interference power over the bandwidth, measured in watt or volt2; and
S/N is the signal-to-noise ratio (SNR) or the carrier-to-noise ratio (CNR) of the communication signal to the Gaussian noise interference expressed as a linear power ratio (not as logarithmic decibels).
So, to increase the capacity (in bits/s) of your Internet connection over a telephony link, you will need to:
DSL link uses both, a increased bandwidth(wide-band) channel and a improved signal/noise rate:
Unlike traditional dial-up modems, which modulate bits into signals in the 300–3400 Hz baseband (voice service), DSL modems modulate frequencies from 4000 Hz to as high as 4 MHz. This frequency band separation enables DSL service and plain old telephone service (POTS) to coexist on the same copper pair facility. Generally, higher bit rate transmissions require a wider frequency band, though the ratio of bit rate to bandwidth are not linear due to significant innovations in digital signal processing and digital modulation methods.
While DSL technology allows much higher transfer rates, it limits the local loop length (the distance between your DSL modem to Telco's DSL termination equipment) to just a few miles, because its signal uses a much wider frequency range and attenuates quickly.
Normal dial-up uses a narrow frequency range, which limits bandwidth to just 56K, however your modem can be miles away the telephone exchange. Moreover, dial-up data signals can travel over several analog or digital telephone networks without trouble, e.g. you can connect one dial-up modem in Africa to another modem in Canada, while DSL signal can only travel a few miles to your telephone exchange.
Just a little more info on the POTS (Plain old Telephone System) mentioned in the accepted answer. There are some very specific specifications for how the phone system works, many of the reasons are archaic but most are still valid.
Look at an old picture of apartment buildings right after phones became popular--the skyline is COVERED with phone lines because every line was dedicated to a single phone (or party line). Soon they came up with a simple, cheap way to compress 24 lines onto a single digital T1 line. This line was the basis of much of american telecommunications for decades. It's spec'd so that a very simple repeater mechanism can be in-lined to allow cables to traverse the ocean without adding power, and they were made to be very easy to multiplex/demultiplex.
The T1 line, being digital, has a Very Specific bandwidth that cannot be changed without changing the internal digital format (which makes it no longer a T1 line and breaks ALL the hardware that supports it currently). When it's split out into phone lines, what you get is a portion of a digital signal interpreted as analog. You will not be able to exceed the original digital bandwidth--you'd be lucky to come anywhere near it considering the conversion to/from analog.
Consider yourself lucky though, some of us spent years on 110/300 baud dial-up (And we were happy to have it!) Actually it was really exciting when our MUD based BBS upgraded to 1200 and we could see the results of our attack BEFORE starting to type in the next command.
The introduction to these Wikipedia pages give you the answer:
Essentially, the xDSL technology uses extra frequency ranges that are not normally used for voice, which is what 56K (and under) dial-up was using.
This is done by using filters on each side of the line so as to split the frequency ranges between traditional audio and the other frequency range (higher frequencies) for the ADSL technology. Because of this, this requires special equipment at the exchange (possibly because the frequency range use for voice wouldn't propagate to the same distances).
EDIT: Note that some ISPs include digital cable/fibre under the term "broadband" (probably to simplify, for commercial reasons): in this case, the signal may be digital all the way through. The range and speed of fibre optics will be better than copper-based lines that use the voice range of frequency (used for old dial-up), but the technology is completely different.
Dialup is slower because it uses much less bandwidth than DSL. A modem uses only 4 kHz of the available spectrum, while DSL can use up to 4 MHz, which is 1000 more. DSL also uses more sophisticated modulation techniques.
Here's an answer that doesn't involve much information theory or technical terminology:
Devices, be they telephones or modems, communicate over the phone lines by sending electricity down the line. The information is encoded by changing the levels of electricity on the wire. On a voice line, those changing levels correspond to the noises you're making into the microphone.
Anything that communicates on a wire, from a telegraph to a 1 Gbits/s Ethernet cable, in the end, communicates by putting pulses of electricity on the wire that the other end can detect.
The more information you want to send down the wire, the faster you have to vary the electrical signals. Morse code involves just a few changes per second, a voice conversation can involve changing the signals thousands of times per second, and high-speed Ethernet can involve tens of millions of changes per second.
The more changes per second you have, the more difficult the circuitry in between has to be, and the better-shielded the wires have to be, as miscellaneous transient disruptions cause more problems on higher-frequency signals.
When the telephone system was originally put together in the late 19th / early 20th century, the first question asked was, how good do we have to be? It was determined that as long as you're able to handle at least 6800 changes per second, (a signal up to 3400 Hz), audible audio will come through, though it will seem a bit 'stunted' - which is why telephones don't sound the same as regular conversation. This worked fine for a hundred years or so.
As computers became popular, people started using modems that made sounds on the line that corresponded to ones and zeroes, but the sounds had to correspond to the range of frequencies in the human voice, limiting them to a few kbits/s. As things improved, they eventually hit the limit of what a phone line can transmit; that limit is about 32 kbits/s, but a simple hack was quickly put in place to bump that up to 56 kbits/s.
About that time, people also realized that you could use a short run of telephone cable to send much higher-frequency signals - up to a few miles when everything worked correctly, but certainly not the tens of miles that a regular phone signal could travel. By having special equipment at the phone company's end, and a DSL modem at the subscriber's end, they could send these special high-frequency signals down 'the last mile' over phone lines that were never really intended for them.
Dial-up is slow because all of the information you send has to be converted into audio data that can be sent over a standard phone line. Do you live in a rural community? Are you stuck with a turtle-slow modem because there is no DSL or highspeed cable in your area?
I seem to remember the original impetus for moving away from dial-up involved the FCC explicitly limiting the POTS bandwidth to 53kbps (that was later removed) so there was no sense in using more than a 56k modem... then of course when phone lines were digitized and multiplexed, and you didn't have a circuit switched pathway you would then loose the ability to perform modulation tricks and harmonic sampling etc to pull out higher virtual clock speed.
