Invention is simply deciding what you need to say. This includes choices about what types of facts must be included and what types of arguments would be most effective, depending on your audience. In modern writing, this is called the brainstorming stage. Aristotle famously said this about this stage: that you must "discover the best available means of persuasion."
Arrangement is concerned with choosing the best organization for your argument. The Greeks had a very strict structure for arrangement, but modern writers have a lot more freedom. Arrangement might involve outlining and choosing what order in which to present facts and examples. Introductory paragraphs and conclusions are part of the arrangement of a modern paper.
In classical rhetoric, style refers to two different types of emphasis a writer or speaker could use to make his work more appealing: ornamentation and orchestration. Ornamentation means imagery, like using metaphors and similes. Orchestration has to do with the rhythm of the language. In modern writing, style is sometimes referred to as a writer's voice.
Memory was a very important aspect of ancient Greek and Roman rhetoric, but is not as much in use today. Speakers would often compose their speeches entirely in their heads, so they used mnemonic devices and strict patterns to make sure they remembered their whole speech.
Delivery is the act of presenting the speech. When planning the delivery of a speech, the Greeks and Romans would leave nothing to chance. They planned out arm movements, eye contact, posture -- everything. They also planned the tone of their speech, how fast or slow to talk, the rhythm of their speaking and the volume of their voice. The idea was to make an emotional connection with the audience, much the way an effective actor can move an audience to tears or laughter.