Yes, logs would be a classic example of an append operation. Email is implementation dependent. For example, Maildir format email stores are one email per file, so no appending needs to take place. With mbox format, all the emails are in a single file, so new ones get added on the end - they are appended.
A database update would be an example of a random write type operation, for example if you decided to update the Firstname of everyone in a database, this would result in a series of writes to the middle of the file that contains the database.
What Google are saying here is, to keep performance optimal and if you can get away with it, then add new information to the end of existing files rather than modifying the content already in them. If you need to delete data from a file, then just flag it somehow and delete it later when performance isn't important, like in a cleanup batch job. An example of this would be with mbox format email stores again. If you delete an email, this is just tracked in a separate file. Then later you can "compact" the file, which effectively rewrites the file without the deleted emails.