I understand that most ordinary people were illiterate back then, so few would be able to write about their lives themselves, but there must be exceptions, right?
I've read The Tale of Genji, and even if it's not written by a Genji himself, it's still interesting to read.
Please exclude religious scriptures. Some would argue that the Bible or Koran are example of real peoples stories, and others would argue that it's the words of God. That's another post.
The Booke of Margery Kempe (although it was dictated) is one of the earliest (if not the first) female autobiography, and definitely the first in English. The Journey through / Description of Wales by Gerald of Wales contains some interesting autobiographical elements (alongside topographical and ethnographical parts). A biography of William Marshal (L'histoire de Guillaume le Marchel) was written shortly after his death and was sourced through his own household, Ulrich von Lichtenstien wrote a, supposedly, autobiographical account of his deeds in arms (Frauendienst, trans. Service of a Lady). There are lots of Latin hagiographies, Vitae (lives), and other retellings of peoples lives which reflect the authors' contemporary concerns as much as they do the life of the individual they are discussing. The Letters of Abelard and Heloise also deserve a mention, although the precise authorship is questionable.
Remember that any biographical, autobiographical, or indeed any text is written for a purpose. I personally enjoy reading the political memoirs of modern politicians (Blair, Bush, Obama) to see how they attempt to sanitize and justify their actions during their political careers (although Dreams of My Father occupies a rather odd position in political memoir).
Ps. define 'normal'!
The Hekanakhet Letters (spelling varies) are a set of letters between a priest and his household that provide a lot of insight into the functioning of what was basically a "middle class" household in ancient Egypt. You can probably find a full text translation of them online. If you're looking for a book, John Ray's "Reflections of Osiris: Lives from Ancient Egypt" is pretty readable, and contains a couple other scenes of everyday, non-royal life.
There are quite a few texts/fragments from Vindolanda that can be quite interesting to look at. Some of them are translated to English on the site, but the more fragmentary they are, the harder it is to get the jest of the latin text. Anyway, these are mostly letters (home or to some other place) from the Roman soldiers at the Vindolanda camp and as such they aren't exactly the stories you are asking for, but rather every-day messages from common people.
I particularly enjoy this letter to a father from a boy in Egypt. The exact book I found it in eludes me at the moment because I just moved and have not finished unpacking, but you can find copies of it online.
The letter is by a boy allegedly named Theon and demonstrates that sons have quite a history of being, well, spoiled little shits, where he whines and complains and threatens his father for not taking him to Alexandria.
He starts the letter off courteously enough, then immediately starts whining, as the only word I can think of.
What amuses me most, I think, is that this person has the unique chance to live on in history, and this is what his name is immortalized by: throwing a tantrum.
"Theon to his father Theon, greetings. A nice thing to do, not taking me with you to the city. If you refuse to take me with you to Alexandria, I shall not write you a letter or speak to you or wish you good health. So: if you go to Alexandria I shall not take your hand or greet you ever again. If you refuse to take me, this is what happens. And my mother said to Archelaos, “He’s upsetting me, take him away!” A nice thing to do, sending me these grand presents, a hill of beans. They put us off the track that day, the 12th, when you sailed. Well then, send for me, I beg you. If you don’t send for me, I shan’t eat, I shan’t drink. There! I pray for your health. [Address] Deliver to Theon from Theonas his son."
An online copy of it can be found here.
Alexander Neckam was a student in London and Paris in the 12th century, and he kept a journal. That journal is excerpted and contextualized in [Urban Tigner Holmes' Daily Living in the 12th Century, based on the observations of Alexander Neckam in London and Paris] (http://books.google.com/books/about/Daily_living_in_the_twelfth_century_base.html?id=NLgYAAAAYAAJ) and it's quite wonderful slice-of-life stuff.
A century earlier you have the autobiography (perhaps the first of the high middle ages) of Guibert of Nogent, an abbot in France - he called his autobiography 'Monodies' after a form of Greek drama. He's pretty thorough about his (often strange) life. Like most work of the middle ages, religion is inescapable but this edition helps contextualize it.
The Tales of Glückel of Hameln is a diary of a Jewish businesswoman who lived between 1646 – 1724. It was at the end of the Medieval age.
She describes how she engaged in trade, ran her own factory, life during the Great Northern War, plague, pirates, soldiers, the false messiah Shabbatai Tzvi, murder, bankruptcy, wedding feasts, births and deaths.
It's written from the perspective of a religious Jewish business woman of the Rhine valley in that era, but it's very much a day-to-day normal life diary (during a particularly turbulent and precarious era).
The Bergen Bryggen rune inscriptions comes to mind.
Historically they're very important. Before they were found in 1955 no one knew if runes were used for more than of names and solemn phrases. Then, boom! 670 rune inscriptions are found, from the 12th century to the 14th Century, and suddenly our understanding of runes and how they were used and by who, and for how long, is changed. The inscriptions are on wood, mostly pine, and some are on bone. It's obvious they are written and used by common people, not just an educated elite.
They're all fairly short messages. Some are markers of property, some short religious formulas, some are busines related and some are personal messages to and from people. They were meant to be read, then thrown away, they were not meant as more than a means of communicating, like an ancient text message if you will. Those are the ones that fit your question the best.
Since I'm like a 12 year old about these things I'll admit that the dirty runes always crack me up. Some of them are so dirty that english translators refused to translate them until recently. (According to the museum now displaying them in Bergen.)
felleg er fuþ sin bylli fuþorglbasm
Lovely is the pussy, may the prick fill it up!
ost min kis mik
My love, kiss me
Gyða segir at þú gakk heim
Gyda tells you to go home (That one reads like something out of a Harek the Horrible comic strip to me.)
Smidr sard Vigdísi
The smith fucked Vigdis
Áli er strodinn i rassin
Åle is fucked in the ass. Åle is a male name, so this is a reference to male on male anal sex.
One of the pieces with runer says that it was owned, written and read by Jón silkifud, Guttormr fudsleikir og Jón fudkúla. Which translates into John silk-pussy, Guttorm pussy-eater and Jon pussy-ball (Whatever a pussy-ball is, that is).
Then there are messages like "I would ask you this, that you leave your party. Cut a letter in runes to Ólafr Hettusveinn's sister. She is in the convent in Bergen. Ask her and your kin for advice when you want to come to terms. You, surely, are less stubborn than the Earl."
This was was written on a wax tablet and was supposed to be hidden under the layer of wax with some trivial writing. It was a top-secret message calling to someone to change the sides in the civil war. Ólafr Hettusveinn is probably Olav Ugjæva, died in 1169.
Anyway, not really the most eloquent messages, but dirty messages from right after the viking age is always worth reading.
Still religious, but I think you'll like reading the autobiography of st. Theresa of Avila. Putting the dogmatic/mystic fragments aside, it's an incredibly rich first account of monastic life in 16th century Spain. Also, since she begins with her childhood, there are a few interesting anecdotes in there.
Emmanuel Le Roy Ladurie's Montaillou, the Promised Land of Error is quite an interesting book from a social-history perspective. It uses inquisition transcripts from 14th Century France to give us an idea of what village life was like in the High Middle Ages. We get to hear something of ordinary people's voices (albeit voices that were written down by inquisitors, rather than the people themselves). A fascinating read.