The 74HC00 family followed, and improved upon, the 74C00 series (which provided an alternative CMOS logic family to the 4000 series but retained the part number scheme and pinouts of the standard 7400 series (especially the 74LS00 series) .
HCT stands for high-speed CMOS with transistor–transistor logic voltages. These devices are similar to the HCMOS types except they will operate at standard TTL power supply voltages and logic input levels. This allows for direct pin-to-pin compatible CMOS replacements to reduce power consumption without loss of speed. 
HCU stands for high-speed CMOS un-buffered. This type of CMOS contains no buffer and is ideal for crystals and other ceramic oscillators needing linearity.