To begin, we must examine the general hierarchy of fashion requirements: The notion of "formal" dress is actually a group of dress codes that are strictly adhered to in specific situations. Generally speaking, dress categories can be constructed as follows: Formal, Semi-formal, Informal (Business Professional), Smart Casual, Business Casual, Casual, and Active.
Formal attire traditional implies "white tie." For western evening dress this is characterized by white and black clothing consisting of a black tailcoat, white bow tie, white waistcoat, and wing collar shirt. (AKA full/evening dress, white tie and tails, or even top hat and tails; This attire was often accompanied by a black top hat.) Morning dress, the formal requirement before 6pm or before dark is a morning coat with waist coat and striped pants. (Often all components are grey.) While these formal garments are somewhat outdated, they are still adhered to in royal events or even weddings. Furthermore, formal attire would include robes or other garments worn by specific professions such as priests or judges.
According to traditional western dress, semi-formal is essentially synonymous with "black tie." Consider an important distinction between evening semi-formal (after 6pm) and morning semi-formal: In the morning, a stroller (black or grey single or double breasted coat with formal pants, grey/silver tie, and grey/black waistcoat) is the standard. This outfit is also known as "black lounge," as it is a variant of the "informal" lounge suit, which would be some other color (black was traditional reserved for formal/semi-formal attire exclusively). In the evening, a tuxedo (or dinner jacket) is the proper apparel. Although in some circles (unofficial or unsophisticated) semi-formal can imply shoddy dress, (typical "middle-school dance semi-formal" attire: khakis, tie, shoes as casual as Sperry's), to the sharp dresser, fashionably informed, or social savant semi-formal has an entirely different implication. *Note that a wedding is an example of a fashionably sophisticated event.
Perhaps the most commonly misused terminology is "informal attire." Many take this at face value to imply casual garments, which could simply be jeans and a t-shirt. Informal is actually more of a subset of formal and requires international business attire, business professional dress, or simply a "suit and tie." You would have a lot more flexibility here in terms of colors and styles than in semi-formal. Additionally, various professions allow for differences in appearance. For instance, artsy industries may allow more flair while some banking industries would all but require additional displays of wealth (cuff links, watches, etc).
The realm of "casual" clothing is much more loosely defined and dependent upon the specific situation than "formal" attire. Smart casual is essentially the upper end of business casual, where nice clothes are displayed and tie and jacket remain optional. This category is generally characterized by desire to impress one's peers without adhering to a formal dress code. Sweaters, sweater vests, and other dressy accessories are often exhibited here. Smart casual presents the fashion forward individual an opportunity to wear trendy, less conservative clothing, and/or promote a thematic style.
Casual and active wear are taken at face value. Active implies workout or maybe competition clothing, and casual is just whatever you want to wear (within the etiquette lines of modesty).
Well... Now that you're far more informed on proper dress, we must add a bit of a disclaimer/warning. First the warning: To effectively master clothing, you must be able to read the situation presented: How formal/rare/special is the occurrence? Who is attending? Where is it being held? (etc) These questions will allow you to prepare for the occasion with class. You should never wear black tie to an informal event, and leave the suit at home when the invite says business casual. Disclaimer: With evening dress and tailcoats fading from style, many would consider a sharp (probably three piece) suit appropriate for semi-formal events. Hence, a tux is, in some circles, considered too dressy for semi-formal. Also, as modern fashion moves away from restrictions to black and white for formal attire, the lines between formal, semi-formal, and informal can be a little blurred.