For example, the New Horizons mission was the fastest object launched from Earth, reaching a launch velocity of 58,500 km/h (36,400 mph). At this speed, it was able to make it to the Moon - which lies at an average distance of about 384,400 km (238,850 mi) from Earth - in just 8 hours and 35 minutes.
However, it was the Helios 2 mission – launched in 1976 to study solar processes - that established the record for highest speed achieved by a spacecraft - 240,000 km/hr (150,000 mph). This was done with the help of a gravity assist, where a spacecraft uses the gravitational force of a large object (like a planet or star) to slingshot around it and pick up a boost in velocity.
But even at this speed, it would still take a whopping 19,000 years to reach Proxima Centauri. Another problem is the fact that the spacecraft that rely on chemical propellants exhaust their fuel very quickly in order to achieve top velocity. Ionic propulsion (aka. the Hall-Effect Thruster) are much more fuel-efficient and achieve maximum velocity more slowly.
One of the first missions to rely on an ion drive was NASA's Deep Space 1 mission, a technology demonstrator that rendezvoused with the asteroid 9969 Braille and comet Borrelly in 1998. DS1 relied on a xenon-powered ion drive that over the course of 20 months managed to reach a velocity of 56,000 km/hr (35,000 miles/hr).
Ion thrusters are therefore more economical than rocket technology, as the thrust per unit mass of propellant (a.k.a. specific impulse) is far higher. But it takes a long time for ion thrusters to accelerate spacecraft to any great speeds, and the maximum velocity it can achieve is dependent on its fuel supply and how much electrical energy it can generate.
At this velocity, it would take a spacecraft over 81,000 years to travel from Earth to Proxima Centauri. Once again, that's a very long time. To put it in perspective, a spacecraft that relied on conventional engines would take over 750 generations to reach Proxima Centauri, while one that used ion engines would take over 3,200 generations.
Now compare that to crewed missions. The Apollo 10 spacecraft, which flew to the Moon without landing in 1969, holds the record for the highest speed attained by a crewed vehicle with 11.08 km/s (39,888 km/h; 24,791 mph). At this speed, it was able to make it to the Moon in just under 2 days and 4 hours.
But to get to the nearest star, it would take roughly 114,800 years (or about 4600 generations). That kind of defeats the purpose of sending a mission to an extrasolar system, doesn't it? By the time the spacecraft reached it and was in a position to send back information, anyone who witnessed the launch would be long dead.