There are various approaches. Beside trying to identify an organism personally, you can send a photo or specimen to a trusted individual for feedback. BugGuide.net is the place to go online for reliable, timely identifications of insects, spiders, and kin; iNaturalist.org and various social media groups will also provide identifications, although not all images will be identified and accuracy is mixed.
If you have a specimen or photograph of a well-studied group, you might try to identify using a formal dichotomous key. Dichotomous keys can be found in field guides, scientific papers, and other publications. For example, here is an online key to the grape species of New England.
If a key is inappropriate, which is often the case if you are trying to identify something with only a photograph, then you will need another type of resource. This is also true for the many groups for which nobody has written a key, or if you cannot find a key. In this case, comparisons against images and verbal descriptions, found online, in books, and in scientific literature, will be most productive. However, just because an image or description from a website or book matches your organism does not mean that this is the correct identification. Often there will be other species for which an image or description would also match, simply because the species are similar, as is often true when trying to identify a plant from the leaves alone. Even experts misidentify at times when confronted with similar species; experience is the only real defense against this pitfall.
Visit the Further Resources page to consult a research guide of introductory references to local organisms.