If you're just getting interested in amateur astronomy, you may read with dismay statements like, "best seen from a dark site," "get away from city lights," and "galaxies cannot be seen under light-polluted skies." Well, I'm here to say you can observe lots of objects from a city.
Galaxies, unfortunately, are not among them. I'm sorry, there's nothing I can do about this. Even wonderful astronomical devices called "light-pollution filters" can't help you observe galaxies under a bright sky. The reason is galaxies emit light at all wavelengths. Filters help with objects — such as emission nebulae and planetary nebulae — that emit light in narrow wavelength ranges. Filters pass those wavelengths and block all the rest.
So, with a nebula (or, light-pollution) filter labeled "narrowband" screwed into your eyepiece, you can observe bright emission nebulae like the Orion Nebula (M42), the Lagoon Nebula (M8), and the North America Nebula (NGC 7000). Likewise, a filter labeled "OIII" will allow you to observe bright planetary nebulae like the Ring Nebula (M57), the Owl Nebula (M97), and the Helix Nebula (NGC 7293) from a city.
But even without a filter, a multitude of celestial objects await city-bound observers. Double stars are perfect targets under light-polluted skies. Through your telescope, you'll detect more about a double star than just the difference in brightness between the two components.
For example, if you take your time, you can identify the two stars' colors. Star colors through a telescope are subjective, because we each process color in our own way, but after a few dozen doubles, your descriptions shouldn't be too far off those given by experienced observers.
Next, turn your attention to star clusters. First, target open star clusters: Go for the naked-eye targets like the Pleiades (M45), the Beehive Cluster (M44), Ptolemy's Cluster (M7), the Tau Canis Majoris Cluster (NGC 2362), the Coma Berenices Cluster (Melotte 111), and h and χ Persei (NGC 869 and NGC 884). These objects can be worthy targets, especially at low power. Look for interesting patterns of stars and close doubles.
But don't stop with clusters you can spot without optical aid. Any star cluster brighter than about 9th magnitude is fair game under a city-lit sky. Try M26, M29, M39, and the Owl Cluster (NGC 457), just to name a few.
Next, move to globular clusters. Look for the central condensations, and try to spot outlying stars in the Hercules Cluster (M13), M4 in Scorpius, M22 in Sagittarius, M15 in Pegasus, and M10 in Ophiuchus. Although globular clusters are all round, they don't all look the same. Try to spot differences in shape, star concentration, and size.
I can't close without mentioning the best objects to view under light-polluted skies — the Moon and the planets. In fact, the Moon (especially between First Quarter and Last Quarter) actually looks better if you have some nearby light around you. The light triggers your daytime vision, which is much better at picking out details.
The naked-eye planets, especially Jupiter and Saturn, are visual treats whether you're under the ultradark sky at Lowell Observatory in Flagstaff, Arizona, or near downtown Chicago. Jupiter's four brightest moons provide a never-ending pattern that's fun to follow night after night. And Saturn's rings? That's a sight you have to share with your family and friends.
For more about city observing, check out Phil Harrington's urban skies.