In most cases, I prefer to use a short alphanumeric code, and then have a lookup table with the expanded text. When necessary I build the enum table in the program dynamically from the database table.
For example, suppose we have a field that is supposed to contain, say, transaction type, and the possible values are Sale, Return, Service, and Layaway. I'd create a transaction type table with code and description, make the codes maybe "SA", "RE", "SV", and "LY", and use the code field as the primary key. Then in each transaction record I'd post that code. This takes less space than an integer key in the record itself and in the index. Exactly how it is processed depends on the database engine but it shouldn't be dramatically less efficient than an integer key. And because it's mnemonic it's very easy to use. You can dump a record and easily see what the values are and likely remember which is which. You can display the codes without translation in user output and the users can make sense of them. Indeed, this can give you a performance gain over integer keys: In many cases the abbreviation is good for the users -- they often want abbreviations to keep displays compact and avoid scrolling -- so you don't need to join on the transaction table to get a translation.
I would definitely NOT store a long text value in every record. Like in this example, I would not want to dispense with the transaction table and store "Layaway". Not only is this inefficient, but it is quite possible that someday the users will say that they want it changed to "Layaway sale", or even some subtle difference like "Lay-away". Then you not only have to update every record in the database, but you have to search through the program for every place this text occurs and change it. Also, the longer the text, the more likely that somewhere along the line a programmer will mis-spell it and create obscure bugs.
Also, having a transaction type table provides a convenient place to store additional information about the transaction type. Never ever ever write code that says "if whatevercode='A' or whatevercode='C' or whatevercode='X' then ..." Whatever it is that makes those three codes somehow different from all other codes, put a field for it in the transaction table and test that field. If you say, "Well, those are all the tax-related codes" or whatever, then fine, create a field called "tax_related" and set it to true or false for each code value as appropriate. Otherwise when someone creates a new transaction type, they have to look through all those if/or lists and figure out which ones this type should be added to and which it shouldn't. I've read plenty of baffling programs where I had to figure out why some logic applied to these three code values but not others, and when you think a fourth value ought to be included in the list, it's very hard to tell whether it is missing because it is really different in some way, or if the programmer made a mistake.
The only type I don't create the translation table is when the list is very short, there is no additional data to keep, and it is clear from the nature of the universe that it is unlikely to ever change so the values can be safely hard-coded. Like true/false or positive/negative/zero or male/female. (And hey, even that last one, obvious as it seems, there are people insisting we now include "transgendered" and the like.)
Some people dogmatically insist that every table have an auto-generated sequential integer key. Such keys are an excellent choice in many cases, but for code lists, I prefer the short alpha key for the reasons stated above.