There are nine possible reasons not to use electricity on Shabbat
Opinions about electricity range from deorita, d'rabanan and technially, not really an issur.
The two most commonly cited sources on the topic are the Chazon Ish and R. Auerbach who's opinions on this vary greatly.
Igniting a fire
The basic example of using electricity, (turning on an incandescent light switch) violates the Torah prohibition of lighting a fire on shabbat. (the metal glows, and gets hot) However, even though this was the first application of electricity, and it's the most common, and clearly not allowed, it is not a catch all reason for all uses of electricity.
Many want to say that using electricity creates situations of Molid (creating something new) However, R. Auerbach says that you can't expand Molid beyond its usage as defined by the Talmud.
The Chazon Ish says that closing a circuit to create current falls under the deoraita of Boneh. Rabbi Shlomo Zalman Auerbach disagreed in the strongest terms. The reasons for this require an indepth understanding of Boneh.
Some try to argue that turning on or off an electrical appliance, or just using a button on it, is the "final step" that finishes the the item. R. Auerbach greatly dissagreed with this.
If the electrical device always and purposefully creates sparks, (like my radiator for example) the creation of sparks might be a problem.
Additional fuel consumption
In Israel, or places where power plants are run/owned by Jews, the use of electricity might cause these power plants to consume more fuel.
Electricity entering into a wire might heat that wire (according to the Chazon Ish). However, in recent years the prevalence of solid-state technology has made the reality underlying this concern obsolete in many cases.
Rabbi Auerbach says that outside the case of an incandescent light bulb, the true reason not to use electricity on shabbat/yom tov is minhag.
Not Shabastik/ In the spirit of Shabbat / yom tov.
Many say that while electricity in most cases might actually be allowed, it's not in the mood of shabbos and is a "davar chol", something that is normally done during the week. Especially after the past 80 years of Jews not using electricity on Shabbat.