DSP algorithm implementations (software coding) are easiest when your microprocessor has an FPU hardware, but it's also possible with integer (fixed-point) ALUs too.
This means that your microprocessor should either have a dedicated FPU, or it can use a (very inefficient) software emulation for FPU operations. Alternatively one can try a more difficult to program, but more efficient to run, fixed-point approach with integer based arithmetic as well.
Some legacy microprocessors (such as intel 80386,80486, motorola 68010,68020) were based on integer ALU only, and their FPU unit were sold in a separate optional co-processor (80387,80487 or 68881).
Nowadays most (but not all) CPUs come with some 32-bit (or even 64-bit) FPU hardware built-in. But that's not enough.
DSP algorithms require not only an capable arithmetic unit, but also high data throughputs depending on the application. A speech coder can very well be impemented in real-time with a mediocre CPU, while a video encoder in real-time may be hard to achieve.
Real-time means that your processor should be capable of finishing its chunk of job, while live data (opposed to offline data) is fed into it, without overflowing internal storage buffers. This happens, for example, when you play an electronic keyboard; as you press the keys, their sound should be immediately (less than detectable delay threshold) processed and output, before the next keys were pressed (or they will overwrite and skip previously pressed key notes). Or, you want to record input audio as an mp3 file. The input data stream through microphone should be encoded into MP3 audio format before the next buffer filled in so that previous data buffer won't be overwritten.
For high data-rate real-time applications, general purpose CPU may be inefficient or insufficient due to its architecture; so the architecture of CPU is modified. SIMD based FPUs, more cores for parallel processing, optimized caches, and special RISC instruction set (as opposed to CISC) should be employed.
And that's what is referred to as a special purpose DSP processor.