To illustrate this problem, Woods Hole Sea Grant created a poster detailing how long trash remains in the environment, and where the trash is coming from.
This is one reason why recycling is so important: it keeps trash like this out of the environment. Once trash items enter the environment, it can take a while until they go away. While time works its toll on these items, they don't stay in one place. In the ocean, this results in the trash being moved around by ocean currents.
Image from the NOAA Marine Debris Program
In the Pacific Ocean, this has created what is known as the "Great Pacific Garbage Patch." In the above picture, the area marked "Convergence Zone" is where much debris accumulates. You can't actually see much of the patch with the naked eye, as it's composed of bits of trash (mainly plastic), and there are separate "patches" of garbage within the convergence zone. A good description of what you can see at the garbage patches can be found here.
But not being able to see the plastic does not mean that it isn't causing problems. These tiny bits of plastic enter the found chain, due to being ingested by marine organisms such as fish. The chemical composition of the trash that animals eat can mess with the animals' organ systems: affecting their quality of life and even reproduction rate.
As illustrated in the above photos, the bits of plastic that fish eat can be quite significant, leading to problems. A pressing problem for the fish is a nutritional deficient: if its stomach is full of plastic, it feels full, so it doesn't continue feeding, but it hasn't actually gained the correct amount of nutrients it needs.
Even though the garbage and plastic that makes its way into the ocean is invisible to us, it obviously has an effect on marine organisms.
To wrap this up, the following 4 minute video illustrates the harsh life of the organism that is perhaps most affected by this pollution: the plastic bag.
Gregory M. (2009). Environmental implications of plastic debris in marine settings– entanglement, ingestion, smothering, hangers-on, hitch-hiking and alien invasions. Phil. Trans. R. Soc. B. 364(1526): 2013-2025.
Muncke J. (2009). Environmental Hormones in Food Packaging: Migration into Food and the Environment. PDF.
Pichel, W., J. Churnside, T. Veenstra, D. Foley, K. Friedman, R. Brainard, J. Nicoll, Q. Zheng, and P. Clemente-Colon. (2007). Marine debris collects within the North Pacific Subtropical Convergence Zone. Marine Pollution Bulletin 54: 1207-1211.
Rios L.M., Jones P.R., Moore C., Narayan U.V. (2010). Quantitation of persistent organic pollutants adsorbed on plastic debris from the Northern Pacific Gyre’s “eastern garbage patch.” Journal of Environmental Monitoring. 12(12): 2189-2312. PDF.