I have always used "I reckon" to mean, "I have applied a process of thought and come to this conclusion".
"I think" is a statement of my assumptions.
"I believe" is generally something I cannot prove or defend, specifically referring to my "beliefs" in a religious or spiritual context.
For some perspective, I am a native speaker from the Southern United States.
english.stackexchange.com/users/608/Anthony
"I think" is a statement of cursory conclusion, while "I believe" indicates a more deeply considered and committed position.
Note that someone may say "I think" when they actually mean "I believe", simply because it might be received as less confrontational.
"I reckon" carries an ever lower level of assertion than "I think", (or perhaps an even higher level of conciliation).
english.stackexchange.com/users/460/Chris Noe
As an example... When a sheriff in the deep south says, "I reckon I'm takin' you to jail.", it means he's definitely taking you to jail.
There is no question in that sheriff's mind.
He doesn't think or believe, he knows. ;~)
And an up vote for Anthony.
english.stackexchange.com/users/742/wdypdx22
I'd consider both "I believe" and "I think" to reflect roughly equivalent levels of (limited) confidence.
"I reckon" is significantly lower-confidence- more akin to "I'd guess" than than it is to the other two.
english.stackexchange.com/users/577/Jaydles
I believe "I reckon" is more commonly used in British English.
I think that in American English, it's considered to be colloquial.
english.stackexchange.com/users/189/Paused until further notice.
All three are essentially interchangeable, but many think
I believe reflects uncertainty on the part of the speaker.
english.stackexchange.com/users/298/Chris
I believe (heh-heh) that "I believe" acknowledges the possibility of error, and the shortage of evidence, more explicitly than "I think" does.
"I reckon" originally meant "I calculate" and so carries a far stronger promise of accuracy.
If you begin a sentence with "I reckon", you better be right.
english.stackexchange.com/users/4040/Malvolio
Rather than just pointing out degrees, let's talk about what you can hold a conviction in .
These terms also connote degrees.
You can think a conjecture is true.
You can believe in a theory, holding it to be true.
One of them requires some foundation even if only subjective from the perspective of the speaker.
While the other includes possibilities that may be clearly absurd.
I think these lottery numbers will be lucky.
Raises no questions to me.
I believe these lottery numbers will be lucky.
Makes me wonder what you know about the method of number selection at the lottery.
That's not to say every belief is reasonable.
Most are certainly not reasonable, but they're still not random.
I'm a Christian
: I believe the world was created in 6 days.
You're clearly nuts.
But, I understand why you believe that.
It's in your cannon.
As for reckon it has two meanings.
Meaning to calculate.
This is the archaic definition from which the current usage derives, largely lost in the United States.
A southern regionalism for belief.
Often used sarcastically or pejoratively by outsiders to mock southerners.
I'd stay stay away from reckon entirely.
Unless you're planning a comedy skit.
english.stackexchange.com/users/14451/Evan Carroll
These are the three most common ways to say "I think." (At least, I believe so. I mean, I think so. Um...)  Are there any subtle differences between them? Are there situations where one of the three is more suitable than the others? Can you say you hear a particular form all the time in the streets, while the other two are much less common?
Are there any differences between "I believe" vs "I think" vs "I reckon"?
english.stackexchange.com
<word-choice><verbs><differences><synonyms><writing-style>
english.stackexchange.com/questions/1070
english.stackexchange.com/users/524/GSerg
The distinction between "believe", "think" and "reckon" is likely to be very fine, however, "think" and "reckon" may refer to more cursory conclusions that "I believe", which is a more deeply considered and committed position. It's also possible that "I reckon" carries a lower level of assertion than "I think", although other people would assert that it involved calculation and is therefore more certain than "believe" or "think". The terms are interchangeable and may indicate varying levels of certainty in different speakers. Using "I reckon" can appear uneducated and expose the user to mockery.
The distinction between "believe", "think" and "reckon" is likely to be very fine, however, "think" and "reckon" may refer to more cursory conclusions that "I believe", which is a more deeply considered and committed position. It's also possible that "I reckon" carries a lower level of assertion than "I think", although other people would assert that it involved calculation and is therefore more certain than "believe" or "think".  "I reckon" is more commonly used in British English and is significantly lower-confidence- more akin to "I'd guess" than than it is to the other two. "I believe" is generally something I cannot prove or defend, specifically referring to my "beliefs" in a religious or spiritual context.   "I think" is a statement of cursory conclusion, while "I believe" indicates a more deeply considered and committed position.  "I believe" and "I think" reflect roughly equivalent levels of (limited) confidence.    
"I think" is a statement of cursory conclusion, while "I believe" indicates a more deeply considered and committed position.  "I believe" acknowledges the possibility of error, and the shortage of evidence, more explicitly than "I think" does.  "I believe" is generally something that cannot be proven or defended, specifically referring to my "beliefs" in a religious or spiritual context.  "I reckon" carries an ever lower level of assertion than "I think". "I reckon" is more commonly used in British English. There does not appear to be a clear answer to this question. There is disagreement about which of these three denotes more certainty. Some people might claim that "I believe" is more certain than "I think" or vice versa. Equally, there is no agreement as to whether "I reckon" is a more or less confident statement than the other two.
In the Southern US "I reckon" implies more thought than "I think", and "I believe" refers to religious beliefs. "I think" is a statement of cursory conclusion and  "I reckon" is even more cursory, while "I believe" indicates a more deeply considered position. "I reckon" can be used ironically when the speaker is quite sure about the statement. Both "I believe" and "I think" reflect roughly equivalent levels of (limited) confidence but "I reckon" is significantly lower-confidence- more akin to "I'd guess". "I reckon" is more common in UK English but in US English it sounds colloquial "I believe" acknowledges the possibility of error, and the shortage of evidence, more explicitly than "I think" does but "I reckon" originally meant "I calculate" and so carries a far stronger promise of accuracy. These terms also connote degrees but it depends on context - "I reckon" is often used to mock regional dialects in the US and should otherwise be avoided.
 "I reckon" is more commonly used in British English and is significantly lower-confidence- more akin to "I'd guess" than than it is to the other two. "I believe" is generally something I cannot prove or defend, specifically referring to my "beliefs" in a religious or spiritual context.    "I think" is a statement of cursory conclusion, while "I believe" indicates a more deeply considered and committed position.  "I believe" and "I think" reflect roughly equivalent levels of (limited) confidence.    
"I think" is considered to be less committal and confident than "I believe". Some people would consider that "I believe" demonstrates a certain level of doubt. There is no agreement about whether "I reckon" is stronger or weaker in confidence. It is possible that "I reckon" is more common in British English.