Q:

What is the farthest artificial satellite in orbit in our solar system?

When did we send the farthest satellite and what is its role and name ?

A:

The term for orbits in our solar system around the Sun is Heliocentric.
Closed Heliocentric Orbit
The solar observation probe Ulysses is the furthest artificial satellite around the sun. It's in a highly inclined, elliptical orbit ranging from 1.35 Astronomical Units (AU) to 5.4 AU. It was a joint project by ESA/NASA launched in 1990 and decommissioned in 2009. The mission utilised a gravity assist from Jupiter to achieve its high 79° inclination.
It was the satellite with the highest aphelion in a purely heliocentric regime.
At the time of writing, Hayabusa2 and OSIRIS-REx are both in heliocentric orbits with aphelia of ~1.3 AU. Both are on sample-return mission to different asteroids - 162173 Ryugu and 101955 Bennu respectively.

Planetary Orbits
We have sent many spacecraft to orbit other planets within the solar system, meaning they are technically in orbit around the Sun, but not heliocentric.
The most distant satellite currently in operation is Juno which entered orbit around Jupiter in 2016 and has been making atmospheric, gravitational and magnetic observations of Jupiter since. The Galileo probe also orbited Jupiter from 1995-2003. Jupiter orbits at an average of ~5.2 AU.
Until September 2017, Cassini was the most distant operational satellite in orbit around Saturn. Cassini's objectives included:

Mapping Saturn's rings
Observing the geology of Saturn's moons
Measuring Saturn's magnetosphere
Deploying the Huygens lander onto the moon Titan

Saturn orbits at approximately 9.6 AU.
Escaping Spacecraft
At a distance of 141 AU, Voyager-1 is the most distant man-made object from the Sun. Although also technically in a heliocentric orbit - a hyperbolic one - it would not usually be described as orbiting the sun.
Also escaping the solar system are Voyager-2, Pioneer 10, Pioneer 11 and New Horizons.
Here is good list of all current solar system spacecraft.

