First a straight forward answer to your direct question - to get the most out of the coprocessor, your code should be able to use a large number of threads and should vectorize. How many threads? Well, you have 60 cores (+/- depending on which version you get) and 4 threads per core, with a sweet spot around 2 threads per core on many codes. Sometimes you can get good performance even if you don't use every single core. But vectorization is extremely important; the long (512 byte) vectors are a big source of speed on the coprocessor.
Now, on to programming. The Intel Xeon Phi coprocessor uses two different kinds of programming - offload and native.
In the offload model, you write a program, determine which parts of that code have enough parallelism to make use of the large number of cores on the coprocessor and mark those sections with offload directives. Then inside those offloaded sections, you write the code using some form of parallelism, like OpenMP. (Heterogeneous)
In native code, you do not use any offload directives but, instead, use a -mmic compiler directive. Then you run the code directly on the coprocessor. The code you write will use some form of parallelism, like OpenMP, to make use of the large number of cores the coprocessor has. (Homogeneous)
Another variation on these programming models is to use MPI, often in addition to OpenMP. You can use the offload programming model, in which case, the nodes in you MPI system will be the host nodes in your system. (Hybrid) Alternately, you can use the native programming model, in which case you treat the coprocessor as just another node in your system. (Heterogeneous if host and coprocessors are nodes; homogeneous if only coprocessors are use.)
You may have noticed that nothing I have said implies a separate programming style for the host and coprocessor. There are some optimizations you can make that will keep code written for the coprocessor from running on the processor as well but, in general, the code you write for the coprocessor can also be compiled for and run on the host by just changing the compiler options.
As far as real world apps, see https://software.intel.com/en-us/mic-developer/app-catalogs