The mechanism of an aqueduct is really straightforward, the trouble is building one. You need a water source which is higher than where you want to get the water to, and then lay a gently downward-sloping channel for the water to run through from the source to the destination. The height differential imparts a little bit of water pressure. The flow is kept constant by way of large tanks which pool water at various points along the way; there were also sedimentation tanks to trap sediment that got carried along the aqueduct, and sluices to manage the water supply. Normally the channel would run along ground level on a gentle slope, normally moving around obstacles but sometimes tunneling through them. In cases where the line the engineers wanted the water to flow down was above ground level (like when the path of the aqueduct crosses a valley or gorge), they would raise the aqueduct on a series of arches. These provide the most striking images of aqueducts, though channels on arches make up a minority of the length of aqueducts. Surviving examples include the Aqua Claudia of which there's a never-ending stream of photographs for you to look at. There are also hundreds of instances of siphons: since fluid in an enclosed space transfers water pressure throughout the space it occupies, water will flow through a pipe even if there is a deep bend in it if you have an entry point higher than the exit point. The water at the entry point puts pressure on the water in the bend below it, pushing the water up the slope at the other end of the bend and through the exit point.

Once the aqueduct gets to the destination it's really up to you how you want to present it: there is water pressure you could use for [fountains spurting upwards or, more likely, outwards](https://en.wikipedia.org/wiki/Fountain#/media/File:Ricostruzione_del_giardino_della_casa_dei_vetii_di_pompei_(mostra_al_giardino_di_boboli,_2007_01.JPG) (you can manipulate water pressure a bit with smart use of cisterns and pipes) or it can just be a pipe that opens into a basin. The water flows through pipes underground (as it was for the most part in the city), or in covered channels or (rarely) open-air channels depending on the requirements of the environment, and are adorned or not depending on whether anyone wanted to take the trouble (normally not).

References

L.W. Mays, D. Koutsoyiannis, and A. N. Angelakis, 'A Brief History of Urban Water Supply in the Antiquity', Water Science and Technology: Water Supply, 7 (1), 2007, pp. 7-10.

A. Trevor Hodge, 'Siphons in Roman Aqueducts', Papers of the British School at Rome, Vol. 51, 1983, pp. 174-221.