Call-out charge - this is a charge for coming out to a job irrespective of the time then spent on that job. Time then spent is charged on top of this. The call-out charge takes account of travelling time and administrative and logistical costs which are the same for a job of 5 minutes or 5 hours.
Hourly rate - the charge per hour. Fractions of an hour would usually be charged pro-rata (e.g. hourly rate £30, 20 mins £10, 5 mins £2.50) but parts of an hour may be charged as full hours, or in say 30 minute increments.
First hours rate - effectively the call-out charge and the hourly rate combined. Some customers don't like the idea of the call-out charge (because they don't understand that it is the fairest way of charging) so the first hour rate disguises it.
Second hours rate - never heard of this but presumably the same as the hourly rate, although it's possible that someone might have a higher rate for the second hour compared to subsequent hours - another form of call-out charge.
For example, if my call-out charge is £30 and my hourly rate £30, then a job that takes 1 hour would cost £60 (plus any materials/parts). If the job took 2 hrs, then it would cost £90 and if it took 20 minutes, it would cost £40.
Someone else might say "no call-out charge" but charge £60 for the first hour and £30 for subsequent hours, which is exactly the same as the example above.
Another person might just charge an hourly rate of say £50 with no call-out or first hour rate. They would be slightly cheaper for one hour (£50 to £60) and slightly more for 2 hours (£100 to £90) and much more expensive thereafter.