I don't have a specific problem but I can't seem to get the meaning of these terms. Sometimes it helps to have someone else explain them.
So far I have deferred expense--Initially, we (the business) paid for it but didn't use it yet (prepaid insurance) which is why we need an adjusting entry if any of it has expired by the end of the year.
Deferred revenue--Initially, we (the business) received payment but we hadn't earned it yet (unearned fees), which is why we need an adjusting entry if we have earned by the end of the year.
Accrued expenses--We (the business) have not paid for it, but wwe show it anyway (a/p)
Accrued rrevenues--We (the business) have not received yet but did eaun it (a/r) because it occurred during the year.
Can someone explain them in a different way?

Sure, I'll give it a short.

Actually, the explanation you wrote for deferral is pretty good. Except I'm not sure if you got that from somewhere else and don't understand it, or if you wrote it as your understanding. If the latter, sounds like you've got a decent handle on it.

It helps to get the concept of "recognizing" a revenue or expense. Recognizing it means recording it into the books so that it ends up on the statements. We recognize revenues when they are earned and expenses when they are incurred.

But that doesn't always match when the cash payment happens. So both deferrals and accruals are involving something where the recognition of the revenue or expense is not in the same period as the payment. This happens quite frequently.

It also helps to remember that both the recognition and the payment will take place. The payment part is a normal everyday transaction. The recognition part is the adjusting entry.

For a deferral, the payment happens prior to the recognition. Specifically:
Deferred revenue is when someone else pays the company ahead of earnings, pretty much what you've got there. So if your company sells magazine subscriptions and people pay for a year in advance, you have the money but haven't earned anything yet cause you haven't sent anything yet. So the payment isn't (all) in the same time as the magazines being sent. So if by the end of the year three months has gone by and you've sent out three months of magazines, you've earned now three months' worth of what you got paid.

So the adjusting entry is to recognize that 3 months of earnings. You record the earnings so that it will show on your income statement as having been earned, and you remove it from the unearned account because it's no longer unearned. An unearned is a liability cause you still have an obligation. You have an obligation to send the magazines worth the amount you got paid. But at year-end, you are no longer obligated to 3 months of it, but you are still obligation to the other 9 months, which remains in the unearned account.

Deferred expense is the opposite: you pay for something ahead of time. That's a prepaid, an asset, because you will still have something coming to you in the future that has value. Let's say it's prepaid rent. You pay for six months up front. You haven't incurred the expense yet because you haven't "used" your rental. As each month goes by, you've used the rental space and have then incurred that expense. So the adjusting entry will be to recognize that expense so it shows on the income statement. You will also remove it out of the prepaid because it's no longer "into the future" and has no value to you anymore cause it's used.

Just like the deferred revenue, with a deferred expense, any portion still into the future will remain in the prepaid account. If you expense one month of rent at a time, after one month you will have 5 left in the prepaid account.

An accrual is something where you have already earned the revenue or incurred the expense, but the payment hasn't been made yet. Again note that the revenue/expense is in a different time than the payment. The payment is a normal daily transaction. The adjusting entry is to do the recognition part.

Accrued revenue is when you have already earned something but the customer/client hasn't paid for it yet. You do this all the time -- dr receivables and cr earnings. You're "accruing" when you do that. You have to recognize the earnings because you have earned it and it needs to be on the income statement. That part probably already makes sense if you understand that entry.

The difference there with the adjusting entry is when the month ends between billing periods, or well, there's other things that simply need recognized also but I won't go into that. Let's just say you send out bills on Fridays and record the receivable/revenue at that time. But Dec. 31 fell on a Tuesday. So you have Monday and Tuesday that falls in December... if you record nothing until Friday then you've got two days in January that don't belong, and two days missing from December. The day you send bills doesn't change the fact that you have already earned during those two days.

So the adjusting entry is to make sure those two days get into your books in December where it belongs, because you did earn it. Then of course it goes into the receivable account. The receivable account is actually the "accrued" account because it represents what hasn't been paid yet, which is what an accrual is, something that hasn't been paid yet.

Note that in this case, the adjusting entry is the first entry made, to recognize the revenue. The payment entry comes later. In the case of a deferral the payment entry comes first, and the adjusting entry (or entries) comes later.

Accrued expense. You've incurred the expense because the service has already been performed or something provided to you, but you haven't paid for it yet. You do these entries all the time too, every time you stick something into accounts payable. You're recognizing an expense you've incurred, but you aren't paying it until later. Payables is the accrued account since it represents what you will pay later.

One of the most common accrued expenses is salaries. Similar to the billing date being outside the month, payroll day can be after the month is over. If payroll is on Friday and you normally record it in the books on that day... and Dec 31 again falls on that Tuesday, well, people have worked for you on Monday and Tuesday. Technically, it accrues every day they work, but there's no reason to worry about each individual day. You're concerned that it's the end of the year and you have incurred that expense and need to get it into your income statement.

So you need to get that expense into your books and recognize it. So you estimate salaries for those two days and record the salary expense. Because it hasn't been paid yet, you put it into salaries payable.

Notice I've mentioned that income statement a lot. That's because the entire point of adjusting entries is to recognize certain revenues and expenses. There's various reasons these weren't already recorded & recognized, but it always involves events that span over more than one accounting period... e.g. earnings in one month but the payment in another.

A little summary I like:
Deferral---->$$$$... recognition of revenue/expense
Accrual---->recognition of revenue/expense... $$$$
Showing the order of the recognition and the money. And the recognition part is the adjusting entry.

I hope that wasn't too much explanation. :D