This is a canonical question that will hopefully encompass all of the questions we get asking us to determine what year a bicycle was manufactured (how old is my bike, how old is my frame, etc.). Each "answer" should address a different way to determine the manufacturing year of a bicycle. To some extent, these will also help you narrow down the model as well as it will tell you what distinguishing features to look at.
The span of answers would include things such as:
Cottered cranks vs square tapered or splined cranks and bottom brackets
Cartridge bottom brackets (BB30/90 etc)
Quick releases versus contemporary through-axles
How the steerer attaches to the fork (quill vs. threadless)
Braking system (especially with mountain bikes: v-brake, direct pull, etc.)
Shifting ramps and pins on cassettes/sprockets and chainrings (versus flat sprockets)
Frame material (e.g., shift from tubular steel to use of CF and hydroformed aluminium)
Frame tubing type (e.g. Reynolds 531 tubing introduced around 1934, still used today; Reynolds 501 tubing introduced around 1983). So a frame with an original 501 decal can't be from the 70', 531 decal doesn't help as much because of the range of years produced. Also, see Decals.
*Decals: style/design, amount, placement, color, etc. can help not only determine models, but also year of manufacture. Many changed decal design for the same model bike often yearly, with special/limited editions even more specific. This applies only if decals are original or have been replaced with the identical design. Tubing decals (Reynolds, Columbus, etc.) also changed designs for same tubing made in different years, although the changes aren't made as often, and manufacturers didn't always put tubing Brand decals on every model. Earlier bikes tended to have less fancy, less colorful, less quantity of decals.
Headbadges: most early bikes had actual "badges", often quite detailed and fancy, made of metal, then plastic, and finally just using decals on the headtube. Most bikes lost their metal badges in the 60's and early 70's, although some brands still have actual badges to this day, but in general a real badge indicates an earlier model bike.
Dropout style (including the later inclusion of lawyers lips)
Presence or absence of CPSC reflectors (USA only)
Manufacturing method (shift from lugs to TIG welding, for example)
Number of gears (shift from 2x5 ten-speed to 3x8, 2x9, 1x11, etc.)
Inspection of lug design: pantographs, cut-out, stampings, custom brazing/filing, chrome, fork crow n sweep, etc.
Frame/Fork Braze-ons. Race bikes tended have less braze-ons the earlier the year of manufacture. Most 60's racers and earlier had few or none with cable guides/stops, shifters, bidons, etc. being clamp-on. By the 80's if it was clamp-on, it was a sign of very cheap models and most frames had braze-ons for all cables, with multiple bottle and accessory bosses depending on style and use.
and so forth. Please edit this question if you have an answer that isn't in the index above.
A couple of caveats:
A previous owner could have replaced or upgraded components on a bicycle, so this guide only applies to original equipment
There is a delay between the introduction of a technology (such as indexed shifting) and when manufacturers start selling products that use it. Low-end manufacturers may continue using technology that is severely outdated (such as one-piece cranks, freewheels, or quill stems) because it is cheaper to do so.
A number of bikes are built in a Retro style, but with modern components. Dutch bikes, trendy coffee cruisers, and beach cruisers may be modern but appear to be 1950s styling. These tend to have modern rims and brakes as a giveaway.
Note that bicycles.stackexchange does not do valuations of bicycles. How much your bicycle is worth depends on the location of the buyer and seller, how much the buyer wants the bike, etc. etc. etc.