There are two features in particular which distinguish quantum computers from conventional computers. The first is that quantum computers use quantum bits called "qubits" instead of traditional "bits". While bits are always one of two values (0 or 1), qubits have an indeterminate value until they are measured; they can be 0, 1, or a sort of combination/intermediate value. (Until measured, at which point the qubit takes a value of either 0 or 1.) This is called superposition.
Because qubits can take many states at the same time, qubits can essentially try all possible combinations of states at once whereas traditional bits could only represent one combination per computation cycle. Performing an operation on a quantum system is like performing it simultaneously to all combinations of states. This allows qubits to be much faster than bits for certain types of calculations. The enormous advances promised by quantum computing requires not just superposition, but also another feature of quantum systems called entanglement. Entangled qubits are linked such that they cannot be described independently of one another; their states are connected regardless of the physical separation between them. Entangled qubits are like a superposition of states of different qubits.
When a measurement is made on one qubit (forcing it into a particular state), the measurement also influences the states of the entangled qubits. With superposition and entanglement, the number of operations that a quantum computer can perform with a given number of qubits is exponentially greater than is possible with a traditional computer using the same number of bits. One application is in factoring of large numbers, useful in the fields of encryption and code-breaking.
Quantum computers could also be used for chemical and biological research by enabling simultaneous modeling of many potential configurations of atoms in molecules or folding patterns of proteins instead of iterating through each possibility one after another.
[ This is a good video explaining quantum computing in increasing detail.
Written information above taken primarily from IBM's web and its pages, Deutsche Welle, and ExtremeTech, and HowItWorks. Some also paraphrased from these and other in these sites. ]