Strictly speaking, in school a "period" is defined as a specific time in the day when you have some scheduled school activity. A "class" is defined as a period during which you receive instruction.
Some schools have a short "homeroom" period in which the teacher takes attendance (documents which students are present and which are absent). "Recess" is a short mid-morning break between two classes, that counts as a "period". "Lunch" is a longer, early afternoon period during which the students eat, play, socialize, etc.
In practice, however, "period" and "class" often mean the same thing -- a time during the school day when you receive instruction. Some schools will count the periods differently from others, so your fifth class might be your sixth or seventh period.
Additionally, some schools have students take the same series of classes every day of the week. Other schools have students take different classes on different days, meaning that, on certain days, some students might have a "free period" between classes.
There are no standards. Use the conventions of your school.
As for the second part of your question: "consecutive" and "back-to-back" mean the same thing -- two activities that occur one after the other. "In a row" also means the same thing when talking about activities. However, the word order may be different:
I had two consecutive science classes.
I had two science classes back-to-back.
I had two science classes in a row.
