I thought this question or - one very similar - had come up before, but I can't find the thread now!
Anyway, the essential thing to remember is that LEDs are d.c. devices. As diodes they produce light in the forward direction but unlike "normal" diodes they have a low breakdown voltage in the reverse direction.
Therefore to run them on a.c., unless the voltage is very low, it requires reverse voltage protection for the LED such as a high voltage diode in series to block any reverse current.
Unless the LEDs are of the type which are "constant current" all LEDs require a method of limiting the maximum forward current to safe levels (e.g. a resistor on d.c. circuits; a.c. circuits could use a resistor or a capacitor).
To make a chain which can run on a.c. the series string would be simplest. A parallel circuit would require a low voltage (e.g. from a transformer, so not practical as part of the string) plus individual current limiting resistors or capacitorsfor each LED. Series-parallel circuit combinations could use groups of LEDs each with common current limiting. All configurations would need diode(s) to block reverse voltage, as above.
For the series method only one reverse diode is required. The string is designed so that "umpteen" LEDs whose forward voltage drops (plus the reverse protection diode) in total add up to near the nominal a.c. supply voltage. One overall current limiting resistor or capacitor would suffice for the whole string, calculated to limit the peak current through the LEDs at the peak a.c voltage (1.414 x nominal a.c voltage.)
Hope that helps!