Vtec is a system that changes the cam profile and lift to suit driving conditions. In a simple vtec, the engine switches between two distinct profiles: 1- a low-end torque, economy profile,
2- a high power, high RPM profile
The car usually starts with the economy profile, switching to the power profile when needed.
Then you have advanced VTEC systems. The 7th gen VTi had a 3 stage Vtec, i.e. 2 distinct cam profiles, plus a 12 valve mode.
The SOHC i-vtec is a bit different. The power profile, or "hot cams", are normally engaged. When the ECU detects a low load, cruise condition, it switches to the eco profile.
i-vtec systems are paired with a VTC system. It advances the cam timing when above 4800 RPM and so, providing more power.
Edit: SOHC i-VTEC engines don't have VTC. Only DOHC engines have them. That's why City and Reborn feel a bit weedy at low RPM.