Obituaries are shorter or longer articles announcing a person's death, summarizing their professional and personal achievements, and listing their direct antecedents and surviving family members. The reason why obituaries are such a valuable resource in historical research is obvious -- they contain much more information than, say, death notices, which only report the event and sometimes state basic information about the funeral service. However, these latter documents can also be useful, especially when the research only needs confirmation of the fact/date of death.
It is essential to be aware that not all deaths find a place in the local newspaper's obit or death notice section. This is in fact the first thing to try and establish before embarking on a full-fledged search. You can do this by entering the name of the decedent into the search box of any online obituary database. However, even if the database does not return any results, that does not automatically mean that there is no obituary for this person. Some of these websites are quite extensive but they do not contain every single obituary ever published in a US newspaper.
If you know the date and place of death, your search will be made much easier. With this information, you could use public library resources such as newspaper collections, to find the obituary you need. Alternatively, if what you need is information documenting the death of an ancestor, you can apply for a copy of their death certificate from the health authorities in the state.
In Illinois, death certificates are issued by the Department of Public Health. A lot of these, all issued 50 years ago or more, are publicly available. There is now even an online database of death certificates issued between 1916 and 1950. The information that these documents contain includes the county and city where the death occurred, the address of the decedent, their full name, date and place of birth, age, sex, occupation, and marital status. Death certificates also include the name of the person who supplied the information about the decedent.
Copies of death certificates can be applied for at the Department of Public Health, in person, by mail, or online, and at the office of the county clerk in the county where the death occurred. An alternative source of historical death records would be the Illinois State Archives.
The State Archives has an extensive genealogical research section, which includes several statewide death indexes, spanning a period of almost two centuries -- from 1763 to 1950. Another resource at the Archives is the Chicago Police Department Homicide Record for the period 1870 to 1930, as well as a number of will and probate records, and county death certificate registers. Not all of these resources are available at the same place, however. You can find a full list of the genealogical resources of the Illinois State Archives.
For Illinois there are also extensive online resources to use in an obituary or a death record search. One pretty long list of such resources is available at the Death Indexes Illinois page. A lot of the links are to regional and county collections and indexes, which are particularly useful -- regional repositories are very often a better source of genealogical information as historical societies and libraries focus on local history, that is, their scope is narrower than that of central repositories, but deeper. The genealogical resources list at Death Indexes also includes cemetery records, newspaper obituary archives, and county wills indexes. An alternative source of information is the Illinois Cemetery Inscription search tool.
Public libraries are the places that tend to keep the biggest newspaper collections. The library of the University of Illinois at Urbana-Champaign, for instance, has a huge collection of historical papers, much of it available online for subscribers. You could browse a list of the collections available at the library, and the periods they cover.
Another newspaper-related resource is available as part of the Illinois Newspaper Project. It involves a search tool for a wealth of newspapers, current and past. The project makes use of the newspaper resources at local libraries and historical and genealogical societies, so the digital collection is indeed very extensive. Some of the newspapers in it are digitalized, but some are still available on microfilm only. When you search for a title, the database will give you its name, period of circulation and where it was found, as well as the format of the record.
At county level, there is a perhaps surprising amount of information available via search engines and document collections. For example, Adams county has its own regional obituary search engine. Local resources also include lists of deaths, which, although they don't provide a lot of details, contain one of the most important pieces of information needed for an obituary search -- the date of death.
Regional resources also include historical mortality indexes, as well as cemetery inscription lists, and funeral home obituary collections. In addition, there are probably hundreds of genealogical and historical societies, which have their own collections, some of which may well feature documents that are not available anywhere else.
In short, there are a lot of sources that one can use when searching for an obituary or another death record, and most of them are very useful, even if they can only supply a piece of information and not the whole. Still, piece by piece, you will be able to get the whole picture of an ancestor's life, especially if you don't limit yourself to just a couple of sources but keep an open mind and never miss an opportunity to tap a source that you haven't considered previously.