Isn't It Ironic…
With apologies to Alanis Morissette, most of the examples in her hit song from a decade ago aren't actually ironic.
Irony is a specific writing technique that builds on a gap (or discontinuity, in English Speaker Parlance) between what is said, and what is meant. Irony is an important tool in narration and telling your own story, because it's such a fundamental human experience and response. It isn't always funny – oftentimes, irony is bittersweet, and in a few cases, it's downright painful.
In most recountings of stories, irony and sarcasm are related – sarcasm can be seen as one kind of irony. Sarcasm and verbal irony are often intentional, and are part of an authorial voice. Some examples of verbal irony include "As sharp as a marble" or "almost as bright as a two watt bulb". To make comparative irony work, the audience has to "get the reference". While an Elizabethan might get "As the ruler of naught" as an ironic reference (and even find it to be bawdy), to a modern audience, it has no meaning whatsoever, or has a meaning that has no ironic intention at all. (For those who are curious, "the ruler of naught" means "the man of the house; the principle "slang" term for female genitalia was "naught" in the 16th century, referring to the lack of visible bids, and is the root of the word "naughty" in modern usage…) Conversely, saying "I'm not used to hitting things that are as fluffy as a carbon nuclei" is ironic if you're a particle physicist, but meaningless to most people.
Now, irony in that sense is fairly common usage, but it's barely the surface of what can be told in a story telling context from irony. A much more powerful technique is tragic irony, where the audience knows what's going on, even when the characters in a narration do not. If you're telling stories from a retrospective, this is doable, though it's easier in fiction. The proverbial cliché example of tragic irony is Romeo discovering Juliet in a drugged sleep, believing her to be dead, and killing himself in grief. It's a technique that's been used in fiction since Greek Myth, when Jason didn't change the color of his sails upon returning with Medea and the Golden Fleece, causing his father to throw himself into the harbor to drown.
Dramatic irony relies, again, on the reader having information the character does not. It can be as subtle as an iceberg (like the movie Titanic), or the underlying premise for a long term plot. Dramatic irony is a way to build tension into your writing.
When writing with dramatic irony, it's very important to touch on the discrepancy between what you expected at the time, and what eventually came about. Dramatic irony, in autobiography, is a way to illustrate how you've grown as a person, the lessons you learned, the foolishness you recovered from. It's a very…non-private sort of writing. Dramatic irony is a technique for showing how you've learned and changed over time, and is a valuable tool for building reader sympathy and empathy.
The last sort of irony that gets used a lot in memoirs is situational irony. It's a disparity between an object and its context. For example, finding ash trays next to gasoline pumps, or a wet bar set up in a temperance union hall, or how a certain Nobel Laureate environmental advocate's utilities and resource bills exceed the median income of all of the residents of his county. Situational irony can be used as the wrap up for a story; again, it's a tool that's used to show how what one set of participants in a story thought was different from the actual occurrences.
Situational irony is one of the hardest tools to use well in storytelling about personal experience, because it has to be used sparingly and with subtlety. It's the punch line to a shaggy dog story…and sometimes, the punchline is delivered repeatedly as a series of body and head blows to the reader. Note that on the Al Gore example, we walked carefully around the circle to let the audience make the revelation of who we were talking about.