I hesitated to answer this, because
switch types really are fairly well
known, and 110VAC can be lethal. I'm trusting you to not kill yourself or
your children, or to burn your house down. Please make _sure_ that you are
comfortable with what you are doing, both for personal and fire safety. Ask
among your friends, neighbors, coworkers and relatives to see if there's an
electronics whiz to help you.
If you really want to do this you may want to get your hands on a Popular
Electronics magazine or something similar, and look for books on the
subject.
You may also consider that if you have homeowners insurance and the system
causes any problem resulting in a claim -- even if it is unrelated to your
modifications -- _you_ will be on trial for _your_ design decisions, and the
burden of proof that your modifications didn't cause the problem will be
_yours_. You will be grilled for your knowledge of electric circuits, and
made to justify everything you did. This will be done by a person who has
the expertise in legal examination to make _me_ look bad, and I'm almost an
expert. Your lawyer will want to hire a professional engineer to analyze
what you did, and said PE will charge enough to buy a new, shrink-wrapped,
UL approved, sprinkler-company-supplied relay every hour or less. Your
lawyer will want more.
If there is a little box on the market already that's UL approved for 24V in
and switched 110VAC out it may be cheap insurance to just buy it and be
happy.
--- Enough disclaimer, already ---
Single throw means that it's just an on-off switch, double-throw means that
there's a center contact that will connect to one "end contact" or the other
depending on whether the relay is energized or not. You'll also run into
the term "pole", as in "single pole" or "double pole" switch; this is just
the number of identical switches that are moved by the same lever (or relay
coil). These will also be abbrieviated: SPST (single-pole, single-throw),
DPST, etc.
A switch that turns on the lights in your house is a single-pole,
single-throw switch. A "three-way" switch like you may have in a long hall
or a basement stair, where either switch will turn on the light is a DPDT
switch in a clever arrangement.
Putting 24VDC on a 12VDC relay coil will run a serious risk of burning out
the relay. You'll be asking it to dissipate 4 times the power that it's
used to. You can do one of three things: Get a 24V relay, wire the coils
of two _identical_ 12V relays in series so they'll share the voltage, or
find the resistance of the 12V relay coil (by measurement or by checking
it's specs) and wire a resistor of the same resistance and adiquate power
dissipation in series. The power in the resistor will be 12 volts squared
(144) divided by the resistance, so a 150-ohm coil will require a one-watt
resistor.
Make sure that the relay coil current (voltage divided by resistance) isn't
more than the irrigation controller can supply.
A relay designed for 220VAC will switch 110VAC safely -- switch contacts of
any kind are usually rated for their maximum working voltage, and it's only
when they get enormous that you need to worry about minimum voltages.
Normally I'd tell you to check the current rating as well, but for a bird
bath you're OK (unless you view the Niagra Falls as a mere bird bath, in
which case you need professional help of an entirely different kind than I
offer). To at least pretend to meet wiring standards, though, the relay
contacts should probably be rated to carry at least the current that the
circuit is fused for (you could put your own fuse in the box, however).
Depending on whether you need to have the relay contacts open or closed when
your 24V is on you either need NO (normally open) or NC (normally closed)
contacts. You probably want to get a SPDT relay, which will have both, and
you can choose when you try it out.
Put it all in a metal box that's well grounded to your wiring system. Make
sure to switch the power side of your wiring and not the neutral. Make sure
the relay contacts are rated for the current that the line is _fused_ for,
_not_ what the pump draws. Think about all the ways that the system could
go wrong, and make no one will get hurt as a result (being irritated is OK,
having to call 911 is not).